# Row-Stochastic Markov Kernels, Finite-Step Laws and Paths

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261024-r03-finite-markov-chain-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[1/3, 2/3], [1/2, 1/2]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[1/3, 2/3], [1/2, 1/2]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[1/3, 2/3], [1/2, 1/2]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0001/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 2. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[1/2, 1/2], [3/5, 2/5]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[1/2, 1/2], [3/5, 2/5]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[1/2, 1/2], [3/5, 2/5]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0002/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 3. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[3/5, 2/5], [2/3, 1/3]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[3/5, 2/5], [2/3, 1/3]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[3/5, 2/5], [2/3, 1/3]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0003/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 4. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[2/3, 1/3], [5/7, 2/7]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[2/3, 1/3], [5/7, 2/7]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[2/3, 1/3], [5/7, 2/7]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0004/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 5. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[5/7, 2/7], [3/4, 1/4]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[5/7, 2/7], [3/4, 1/4]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[5/7, 2/7], [3/4, 1/4]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0005/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 6. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[3/4, 1/4], [7/9, 2/9]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[3/4, 1/4], [7/9, 2/9]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[3/4, 1/4], [7/9, 2/9]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0006/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 7. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[7/9, 2/9], [4/5, 1/5]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[7/9, 2/9], [4/5, 1/5]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[7/9, 2/9], [4/5, 1/5]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0007/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 8. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[4/5, 1/5], [9/11, 2/11]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[4/5, 1/5], [9/11, 2/11]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[4/5, 1/5], [9/11, 2/11]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0008/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 9. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[9/11, 2/11], [5/6, 1/6]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[9/11, 2/11], [5/6, 1/6]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[9/11, 2/11], [5/6, 1/6]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0009/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 10. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[5/6, 1/6], [11/13, 2/13]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[5/6, 1/6], [11/13, 2/13]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[5/6, 1/6], [11/13, 2/13]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0010/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 11. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[11/13, 2/13], [6/7, 1/7]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[11/13, 2/13], [6/7, 1/7]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[11/13, 2/13], [6/7, 1/7]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0011/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 12. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[6/7, 1/7], [13/15, 2/15]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[6/7, 1/7], [13/15, 2/15]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[6/7, 1/7], [13/15, 2/15]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0012/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 13. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[13/15, 2/15], [7/8, 1/8]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[13/15, 2/15], [7/8, 1/8]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[13/15, 2/15], [7/8, 1/8]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0013/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 14. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[7/8, 1/8], [15/17, 2/17]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[7/8, 1/8], [15/17, 2/17]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[7/8, 1/8], [15/17, 2/17]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0014/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 15. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[15/17, 2/17], [8/9, 1/9]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[15/17, 2/17], [8/9, 1/9]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[15/17, 2/17], [8/9, 1/9]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0015/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 16. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[8/9, 1/9], [17/19, 2/19]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[8/9, 1/9], [17/19, 2/19]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[8/9, 1/9], [17/19, 2/19]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0016/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 17. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[17/19, 2/19], [9/10, 1/10]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[17/19, 2/19], [9/10, 1/10]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[17/19, 2/19], [9/10, 1/10]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0017/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 18. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[9/10, 1/10], [19/21, 2/21]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[9/10, 1/10], [19/21, 2/21]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[9/10, 1/10], [19/21, 2/21]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0018/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 19. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[19/21, 2/21], [10/11, 1/11]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[19/21, 2/21], [10/11, 1/11]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[19/21, 2/21], [10/11, 1/11]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0019/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 20. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[10/11, 1/11], [21/23, 2/23]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[10/11, 1/11], [21/23, 2/23]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[10/11, 1/11], [21/23, 2/23]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0020/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 21. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[21/23, 2/23], [11/12, 1/12]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[21/23, 2/23], [11/12, 1/12]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[21/23, 2/23], [11/12, 1/12]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0021/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 22. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[11/12, 1/12], [23/25, 2/25]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[11/12, 1/12], [23/25, 2/25]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[11/12, 1/12], [23/25, 2/25]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0022/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 23. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[23/25, 2/25], [12/13, 1/13]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[23/25, 2/25], [12/13, 1/13]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[23/25, 2/25], [12/13, 1/13]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0023/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 24. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[12/13, 1/13], [25/27, 2/27]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[12/13, 1/13], [25/27, 2/27]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[12/13, 1/13], [25/27, 2/27]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0024/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 25. Row-Stochastic Qualification before Any Markov Step

Use one fixed finite time-homogeneous transition table=[[25/27, 2/27], [13/14, 1/14]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).

View answer and explanation(2, [[25/27, 2/27], [13/14, 1/14]], Yes, [1, 1], Yes, rows=current state, columns=next state)

Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[25/27, 2/27], [13/14, 1/14]], Yes, [1, 1], Yes, rows=current state, columns=next state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0025/)

Finite-state Markov chain · qf-mod24-markov-kernel-audit · candidate





## 26. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[1/3, 2/3], [1/2, 1/2]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[1/2, 1/2] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([1/2, 1/2], [[1/3, 2/3], [1/2, 1/2]], [5/12, 7/12])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], [[1/3, 2/3], [1/2, 1/2]], [5/12, 7/12]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0026/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 27. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[1/2, 1/2], [3/5, 2/5]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[2/3, 1/3] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([2/3, 1/3], [[1/2, 1/2], [3/5, 2/5]], [8/15, 7/15])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2/3, 1/3], [[1/2, 1/2], [3/5, 2/5]], [8/15, 7/15]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0027/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 28. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[3/5, 2/5], [2/3, 1/3]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[3/4, 1/4] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([3/4, 1/4], [[3/5, 2/5], [2/3, 1/3]], [37/60, 23/60])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3/4, 1/4], [[3/5, 2/5], [2/3, 1/3]], [37/60, 23/60]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0028/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 29. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[2/3, 1/3], [5/7, 2/7]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[4/5, 1/5] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([4/5, 1/5], [[2/3, 1/3], [5/7, 2/7]], [71/105, 34/105])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4/5, 1/5], [[2/3, 1/3], [5/7, 2/7]], [71/105, 34/105]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0029/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 30. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[5/7, 2/7], [3/4, 1/4]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[5/6, 1/6] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([5/6, 1/6], [[5/7, 2/7], [3/4, 1/4]], [121/168, 47/168])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5/6, 1/6], [[5/7, 2/7], [3/4, 1/4]], [121/168, 47/168]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0030/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 31. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[3/4, 1/4], [7/9, 2/9]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[6/7, 1/7] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([6/7, 1/7], [[3/4, 1/4], [7/9, 2/9]], [95/126, 31/126])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6/7, 1/7], [[3/4, 1/4], [7/9, 2/9]], [95/126, 31/126]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0031/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 32. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[7/9, 2/9], [4/5, 1/5]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[7/8, 1/8] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([7/8, 1/8], [[7/9, 2/9], [4/5, 1/5]], [281/360, 79/360])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7/8, 1/8], [[7/9, 2/9], [4/5, 1/5]], [281/360, 79/360]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0032/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 33. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[4/5, 1/5], [9/11, 2/11]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[8/9, 1/9] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([8/9, 1/9], [[4/5, 1/5], [9/11, 2/11]], [397/495, 98/495])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8/9, 1/9], [[4/5, 1/5], [9/11, 2/11]], [397/495, 98/495]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0033/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 34. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[9/11, 2/11], [5/6, 1/6]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[9/10, 1/10] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([9/10, 1/10], [[9/11, 2/11], [5/6, 1/6]], [541/660, 119/660])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9/10, 1/10], [[9/11, 2/11], [5/6, 1/6]], [541/660, 119/660]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0034/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 35. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[5/6, 1/6], [11/13, 2/13]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[10/11, 1/11] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([10/11, 1/11], [[5/6, 1/6], [11/13, 2/13]], [358/429, 71/429])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10/11, 1/11], [[5/6, 1/6], [11/13, 2/13]], [358/429, 71/429]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0035/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 36. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[11/13, 2/13], [6/7, 1/7]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[11/12, 1/12] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([11/12, 1/12], [[11/13, 2/13], [6/7, 1/7]], [925/1092, 167/1092])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11/12, 1/12], [[11/13, 2/13], [6/7, 1/7]], [925/1092, 167/1092]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0036/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 37. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[6/7, 1/7], [13/15, 2/15]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[12/13, 1/13] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([12/13, 1/13], [[6/7, 1/7], [13/15, 2/15]], [1171/1365, 194/1365])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12/13, 1/13], [[6/7, 1/7], [13/15, 2/15]], [1171/1365, 194/1365]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0037/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 38. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[13/15, 2/15], [7/8, 1/8]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[13/14, 1/14] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([13/14, 1/14], [[13/15, 2/15], [7/8, 1/8]], [1457/1680, 223/1680])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([13/14, 1/14], [[13/15, 2/15], [7/8, 1/8]], [1457/1680, 223/1680]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0038/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 39. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[7/8, 1/8], [15/17, 2/17]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[14/15, 1/15] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([14/15, 1/15], [[7/8, 1/8], [15/17, 2/17]], [893/1020, 127/1020])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([14/15, 1/15], [[7/8, 1/8], [15/17, 2/17]], [893/1020, 127/1020]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0039/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 40. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[15/17, 2/17], [8/9, 1/9]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[15/16, 1/16] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([15/16, 1/16], [[15/17, 2/17], [8/9, 1/9]], [2161/2448, 287/2448])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([15/16, 1/16], [[15/17, 2/17], [8/9, 1/9]], [2161/2448, 287/2448]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0040/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 41. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[8/9, 1/9], [17/19, 2/19]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[16/17, 1/17] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([16/17, 1/17], [[8/9, 1/9], [17/19, 2/19]], [2585/2907, 322/2907])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([16/17, 1/17], [[8/9, 1/9], [17/19, 2/19]], [2585/2907, 322/2907]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0041/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 42. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[17/19, 2/19], [9/10, 1/10]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[17/18, 1/18] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([17/18, 1/18], [[17/19, 2/19], [9/10, 1/10]], [3061/3420, 359/3420])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([17/18, 1/18], [[17/19, 2/19], [9/10, 1/10]], [3061/3420, 359/3420]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0042/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 43. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[9/10, 1/10], [19/21, 2/21]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[18/19, 1/19] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([18/19, 1/19], [[9/10, 1/10], [19/21, 2/21]], [1796/1995, 199/1995])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([18/19, 1/19], [[9/10, 1/10], [19/21, 2/21]], [1796/1995, 199/1995]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0043/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 44. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[19/21, 2/21], [10/11, 1/11]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[19/20, 1/20] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([19/20, 1/20], [[19/21, 2/21], [10/11, 1/11]], [4181/4620, 439/4620])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([19/20, 1/20], [[19/21, 2/21], [10/11, 1/11]], [4181/4620, 439/4620]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0044/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 45. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[10/11, 1/11], [21/23, 2/23]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[20/21, 1/21] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([20/21, 1/21], [[10/11, 1/11], [21/23, 2/23]], [4831/5313, 482/5313])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([20/21, 1/21], [[10/11, 1/11], [21/23, 2/23]], [4831/5313, 482/5313]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0045/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 46. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[21/23, 2/23], [11/12, 1/12]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[21/22, 1/22] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([21/22, 1/22], [[21/23, 2/23], [11/12, 1/12]], [5545/6072, 527/6072])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([21/22, 1/22], [[21/23, 2/23], [11/12, 1/12]], [5545/6072, 527/6072]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0046/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 47. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[11/12, 1/12], [23/25, 2/25]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[22/23, 1/23] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([22/23, 1/23], [[11/12, 1/12], [23/25, 2/25]], [3163/3450, 287/3450])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([22/23, 1/23], [[11/12, 1/12], [23/25, 2/25]], [3163/3450, 287/3450]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0047/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 48. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[23/25, 2/25], [12/13, 1/13]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[23/24, 1/24] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([23/24, 1/24], [[23/25, 2/25], [12/13, 1/13]], [7177/7800, 623/7800])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([23/24, 1/24], [[23/25, 2/25], [12/13, 1/13]], [7177/7800, 623/7800]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0048/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 49. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[12/13, 1/13], [25/27, 2/27]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[24/25, 1/25] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([24/25, 1/25], [[12/13, 1/13], [25/27, 2/27]], [8101/8775, 674/8775])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([24/25, 1/25], [[12/13, 1/13], [25/27, 2/27]], [8101/8775, 674/8775]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0049/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 50. A Row Initial Law Advances by mu*P

Use one fixed finite time-homogeneous transition table=[[25/27, 2/27], [13/14, 1/14]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[25/26, 1/26] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].

View answer and explanation([25/26, 1/26], [[25/27, 2/27], [13/14, 1/14]], [9101/9828, 727/9828])

Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([25/26, 1/26], [[25/27, 2/27], [13/14, 1/14]], [9101/9828, 727/9828]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0050/)

Finite-state Markov chain · qf-mod24-markov-one-step-law · candidate





## 51. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[1/3, 2/3], [1/2, 1/2]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [2/9, 1/3], 5/9)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [2/9, 1/3], 5/9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0051/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 52. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[1/2, 1/2], [3/5, 2/5]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [3/10, 6/25], 27/50)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [3/10, 6/25], 27/50).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0052/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 53. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[3/5, 2/5], [2/3, 1/3]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [6/25, 2/15], 28/75)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [6/25, 2/15], 28/75).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0053/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 54. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[2/3, 1/3], [5/7, 2/7]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [10/21, 10/49], 100/147)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [10/21, 10/49], 100/147).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0054/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 55. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[5/7, 2/7], [3/4, 1/4]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [10/49, 1/14], 27/98)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [10/49, 1/14], 27/98).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0055/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 56. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[3/4, 1/4], [7/9, 2/9]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [7/12, 14/81], 245/324)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [7/12, 14/81], 245/324).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0056/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 57. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[7/9, 2/9], [4/5, 1/5]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [14/81, 2/45], 88/405)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [14/81, 2/45], 88/405).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0057/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 58. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[4/5, 1/5], [9/11, 2/11]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [36/55, 18/121], 486/605)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [36/55, 18/121], 486/605).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0058/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 59. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[9/11, 2/11], [5/6, 1/6]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [18/121, 1/33], 65/363)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [18/121, 1/33], 65/363).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0059/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 60. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[5/6, 1/6], [11/13, 2/13]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [55/78, 22/169], 847/1014)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [55/78, 22/169], 847/1014).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0060/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 61. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[11/13, 2/13], [6/7, 1/7]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [22/169, 2/91], 180/1183)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [22/169, 2/91], 180/1183).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0061/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 62. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[6/7, 1/7], [13/15, 2/15]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [26/35, 26/225], 1352/1575)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [26/35, 26/225], 1352/1575).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0062/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 63. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[13/15, 2/15], [7/8, 1/8]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [26/225, 1/60], 119/900)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [26/225, 1/60], 119/900).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0063/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 64. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[7/8, 1/8], [15/17, 2/17]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [105/136, 30/289], 2025/2312)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [105/136, 30/289], 2025/2312).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0064/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 65. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[15/17, 2/17], [8/9, 1/9]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [30/289, 2/153], 304/2601)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [30/289, 2/153], 304/2601).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0065/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 66. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[8/9, 1/9], [17/19, 2/19]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [136/171, 34/361], 2890/3249)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [136/171, 34/361], 2890/3249).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0066/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 67. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[17/19, 2/19], [9/10, 1/10]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [34/361, 1/95], 189/1805)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [34/361, 1/95], 189/1805).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0067/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 68. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[9/10, 1/10], [19/21, 2/21]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [57/70, 38/441], 3971/4410)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [57/70, 38/441], 3971/4410).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0068/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 69. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[19/21, 2/21], [10/11, 1/11]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [38/441, 2/231], 460/4851)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [38/441, 2/231], 460/4851).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0069/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 70. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[10/11, 1/11], [21/23, 2/23]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [210/253, 42/529], 5292/5819)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [210/253, 42/529], 5292/5819).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0070/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 71. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[21/23, 2/23], [11/12, 1/12]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [42/529, 1/138], 275/3174)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [42/529, 1/138], 275/3174).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0071/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 72. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[11/12, 1/12], [23/25, 2/25]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [253/300, 46/625], 6877/7500)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [253/300, 46/625], 6877/7500).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0072/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 73. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[23/25, 2/25], [12/13, 1/13]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [46/625, 2/325], 648/8125)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [46/625, 2/325], 648/8125).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0073/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 74. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[12/13, 1/13], [25/27, 2/27]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(1, 0, [100/117, 50/729], 8750/9477)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [100/117, 50/729], 8750/9477).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0074/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 75. Two-Step Probability by Summing Every Intermediate State

Use one fixed finite time-homogeneous transition table=[[25/27, 2/27], [13/14, 1/14]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

View answer and explanation(0, 1, [50/729, 1/189], 377/5103)

Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [50/729, 1/189], 377/5103).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0075/)

Finite-state Markov chain · qf-mod24-markov-two-step-entry · candidate





## 76. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[1/3, 2/3], [1/2, 1/2]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[1/2, 1/2] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [1/2, 2/3, 1/2], 1/6)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [1/2, 2/3, 1/2], 1/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0076/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 77. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[1/2, 1/2], [3/5, 2/5]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[2/3, 1/3] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/3, 3/5, 1/2], 1/10)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/3, 3/5, 1/2], 1/10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0077/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 78. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[3/5, 2/5], [2/3, 1/3]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[3/4, 1/4] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [3/4, 2/5, 2/3], 1/5)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [3/4, 2/5, 2/3], 1/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0078/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 79. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[2/3, 1/3], [5/7, 2/7]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[4/5, 1/5] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/5, 5/7, 1/3], 1/21)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/5, 5/7, 1/3], 1/21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0079/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 80. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[5/7, 2/7], [3/4, 1/4]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[5/6, 1/6] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [5/6, 2/7, 3/4], 5/28)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [5/6, 2/7, 3/4], 5/28).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0080/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 81. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[3/4, 1/4], [7/9, 2/9]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[6/7, 1/7] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/7, 7/9, 1/4], 1/36)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/7, 7/9, 1/4], 1/36).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0081/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 82. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[7/9, 2/9], [4/5, 1/5]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[7/8, 1/8] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [7/8, 2/9, 4/5], 7/45)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [7/8, 2/9, 4/5], 7/45).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0082/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 83. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[4/5, 1/5], [9/11, 2/11]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[8/9, 1/9] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/9, 9/11, 1/5], 1/55)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/9, 9/11, 1/5], 1/55).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0083/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 84. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[9/11, 2/11], [5/6, 1/6]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[9/10, 1/10] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [9/10, 2/11, 5/6], 3/22)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [9/10, 2/11, 5/6], 3/22).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0084/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 85. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[5/6, 1/6], [11/13, 2/13]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[10/11, 1/11] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/11, 11/13, 1/6], 1/78)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/11, 11/13, 1/6], 1/78).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0085/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 86. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[11/13, 2/13], [6/7, 1/7]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[11/12, 1/12] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [11/12, 2/13, 6/7], 11/91)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [11/12, 2/13, 6/7], 11/91).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0086/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 87. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[6/7, 1/7], [13/15, 2/15]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[12/13, 1/13] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/13, 13/15, 1/7], 1/105)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/13, 13/15, 1/7], 1/105).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0087/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 88. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[13/15, 2/15], [7/8, 1/8]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[13/14, 1/14] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [13/14, 2/15, 7/8], 13/120)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [13/14, 2/15, 7/8], 13/120).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0088/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 89. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[7/8, 1/8], [15/17, 2/17]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[14/15, 1/15] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/15, 15/17, 1/8], 1/136)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/15, 15/17, 1/8], 1/136).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0089/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 90. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[15/17, 2/17], [8/9, 1/9]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[15/16, 1/16] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [15/16, 2/17, 8/9], 5/51)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [15/16, 2/17, 8/9], 5/51).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0090/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 91. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[8/9, 1/9], [17/19, 2/19]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[16/17, 1/17] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/17, 17/19, 1/9], 1/171)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/17, 17/19, 1/9], 1/171).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0091/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 92. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[17/19, 2/19], [9/10, 1/10]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[17/18, 1/18] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [17/18, 2/19, 9/10], 17/190)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [17/18, 2/19, 9/10], 17/190).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0092/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 93. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[9/10, 1/10], [19/21, 2/21]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[18/19, 1/19] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/19, 19/21, 1/10], 1/210)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/19, 19/21, 1/10], 1/210).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0093/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 94. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[19/21, 2/21], [10/11, 1/11]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[19/20, 1/20] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [19/20, 2/21, 10/11], 19/231)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [19/20, 2/21, 10/11], 19/231).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0094/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 95. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[10/11, 1/11], [21/23, 2/23]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[20/21, 1/21] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/21, 21/23, 1/11], 1/253)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/21, 21/23, 1/11], 1/253).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0095/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 96. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[21/23, 2/23], [11/12, 1/12]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[21/22, 1/22] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [21/22, 2/23, 11/12], 7/92)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [21/22, 2/23, 11/12], 7/92).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0096/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 97. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[11/12, 1/12], [23/25, 2/25]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[22/23, 1/23] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/23, 23/25, 1/12], 1/300)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/23, 23/25, 1/12], 1/300).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0097/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 98. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[23/25, 2/25], [12/13, 1/13]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[23/24, 1/24] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [23/24, 2/25, 12/13], 23/325)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [23/24, 2/25, 12/13], 23/325).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0098/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 99. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[12/13, 1/13], [25/27, 2/27]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[24/25, 1/25] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([1, 0, 1], [1/25, 25/27, 1/13], 1/351)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/25, 25/27, 1/13], 1/351).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0099/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate





## 100. A Finite Path Includes Its Initial-State Mass

Use one fixed finite time-homogeneous transition table=[[25/27, 2/27], [13/14, 1/14]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[25/26, 1/26] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

View answer and explanation([0, 1, 0], [25/26, 2/27, 13/14], 25/378)

Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [25/26, 2/27, 13/14], 25/378).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0100/)

Finite-state Markov chain · qf-mod24-markov-path-probability · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261024-r03-finite-markov-chain-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261024-r03-finite-markov-chain-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261024-r03-finite-markov-chain-foundations/validation-report.json)