# Exact Finite-Step Markov Powers, Endpoint Joints and Conditional 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-20261025-r06-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. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=1. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 1, [[1/5, 4/5], [1/4, 3/4]], [19/80, 61/80])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 1, [[1/5, 4/5], [1/4, 3/4]], [19/80, 61/80]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0001/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 2. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=2. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 2, [[31/100, 69/100], [23/80, 57/80]], [469/1600, 1131/1600])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 2, [[31/100, 69/100], [23/80, 57/80]], [469/1600, 1131/1600]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0002/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 3. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=3. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 3, [[411/1000, 589/1000], [589/1600, 1011/1600]], [12123/32000, 19877/32000])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 3, [[411/1000, 589/1000], [589/1600, 1011/1600]], [12123/32000, 19877/32000]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0003/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 4. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=1. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 1, [[4/5, 1/5], [1/4, 3/4]], [31/80, 49/80])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 1, [[4/5, 1/5], [1/4, 3/4]], [31/80, 49/80]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0004/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 5. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=2. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 2, [[11/25, 14/25], [7/20, 13/20]], [149/400, 251/400])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 2, [[11/25, 14/25], [7/20, 13/20]], [149/400, 251/400]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0005/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 6. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=3. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 3, [[227/500, 273/500], [91/200, 109/200]], [1819/4000, 2181/4000])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 3, [[227/500, 273/500], [91/200, 109/200]], [1819/4000, 2181/4000]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0006/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 7. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=1. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 1, [[3/5, 2/5], [1/2, 1/2]], [21/40, 19/40])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 1, [[3/5, 2/5], [1/2, 1/2]], [21/40, 19/40]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0007/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 8. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=2. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 2, [[37/50, 13/50], [13/20, 7/20]], [269/400, 131/400])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 2, [[37/50, 13/50], [13/20, 7/20]], [269/400, 131/400]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0008/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 9. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=3. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 3, [[199/500, 301/500], [903/1600, 697/1600]], [16729/32000, 15271/32000])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 3, [[199/500, 301/500], [903/1600, 697/1600]], [16729/32000, 15271/32000]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0009/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 10. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=1. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 1, [[2/5, 3/5], [3/4, 1/4]], [53/80, 27/80])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 1, [[2/5, 3/5], [3/4, 1/4]], [53/80, 27/80]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0010/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 11. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=2. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 2, [[33/50, 17/50], [51/80, 29/80]], [1029/1600, 571/1600])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 2, [[33/50, 17/50], [51/80, 29/80]], [1029/1600, 571/1600]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0011/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 12. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/4, 3/4] to be nonnegative with exact total 1; steps=3. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/4, 3/4], 3, [[1579/2000, 421/2000], [1263/1600, 337/1600]], [25261/32000, 6739/32000])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 3, [[1579/2000, 421/2000], [1263/1600, 337/1600]], [25261/32000, 6739/32000]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0012/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 13. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=1. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 1, [[1/5, 4/5], [1/4, 3/4]], [9/40, 31/40])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 1, [[1/5, 4/5], [1/4, 3/4]], [9/40, 31/40]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0013/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 14. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=2. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 2, [[31/100, 69/100], [23/80, 57/80]], [239/800, 561/800])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 2, [[31/100, 69/100], [23/80, 57/80]], [239/800, 561/800]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0014/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 15. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=3. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 3, [[411/1000, 589/1000], [589/1600, 1011/1600]], [6233/16000, 9767/16000])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 3, [[411/1000, 589/1000], [589/1600, 1011/1600]], [6233/16000, 9767/16000]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0015/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 16. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=1. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 1, [[4/5, 1/5], [1/4, 3/4]], [21/40, 19/40])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 1, [[4/5, 1/5], [1/4, 3/4]], [21/40, 19/40]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0016/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 17. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=2. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 2, [[11/25, 14/25], [7/20, 13/20]], [79/200, 121/200])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 2, [[11/25, 14/25], [7/20, 13/20]], [79/200, 121/200]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0017/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 18. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=3. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 3, [[227/500, 273/500], [91/200, 109/200]], [909/2000, 1091/2000])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 3, [[227/500, 273/500], [91/200, 109/200]], [909/2000, 1091/2000]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0018/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 19. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=1. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 1, [[3/5, 2/5], [1/2, 1/2]], [11/20, 9/20])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 1, [[3/5, 2/5], [1/2, 1/2]], [11/20, 9/20]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0019/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 20. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=2. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 2, [[37/50, 13/50], [13/20, 7/20]], [139/200, 61/200])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 2, [[37/50, 13/50], [13/20, 7/20]], [139/200, 61/200]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0020/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 21. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=3. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 3, [[199/500, 301/500], [903/1600, 697/1600]], [7699/16000, 8301/16000])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 3, [[199/500, 301/500], [903/1600, 697/1600]], [7699/16000, 8301/16000]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0021/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 22. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=1. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 1, [[2/5, 3/5], [3/4, 1/4]], [23/40, 17/40])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 1, [[2/5, 3/5], [3/4, 1/4]], [23/40, 17/40]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0022/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 23. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=2. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 2, [[33/50, 17/50], [51/80, 29/80]], [519/800, 281/800])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 2, [[33/50, 17/50], [51/80, 29/80]], [519/800, 281/800]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0023/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 24. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[1/2, 1/2] to be nonnegative with exact total 1; steps=3. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([1/2, 1/2], 3, [[1579/2000, 421/2000], [1263/1600, 337/1600]], [12631/16000, 3369/16000])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 3, [[1579/2000, 421/2000], [1263/1600, 337/1600]], [12631/16000, 3369/16000]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0024/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 25. An Exact n-Step Row Law, Including Zero Steps

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. Require initial law=[3/4, 1/4] to be nonnegative with exact total 1; steps=1. Give (initial law,steps,P^steps,initial*P^steps).

View answer and explanation([3/4, 1/4], 1, [[1/5, 4/5], [1/4, 3/4]], [17/80, 63/80])

Compute the finite matrix power exactly. At zero steps the identity leaves the initial law unchanged. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3/4, 1/4], 1, [[1/5, 4/5], [1/4, 3/4]], [17/80, 63/80]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0025/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-law · candidate





## 26. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 1, [[1/5, 4/5], [1/4, 3/4]], 4/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0026/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 27. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 1, [[2/5, 3/5], [1/4, 3/4]], 1/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0027/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 28. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 1, [[3/5, 2/5], [1/4, 3/4]], 2/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0028/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 29. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 1, [[4/5, 1/5], [1/4, 3/4]], 1/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0029/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 30. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 1, [[1/5, 4/5], [1/2, 1/2]], 4/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0030/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 31. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 1, [[2/5, 3/5], [1/2, 1/2]], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0031/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 32. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 1, [[3/5, 2/5], [1/2, 1/2]], 2/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0032/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 33. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 1, [[4/5, 1/5], [1/2, 1/2]], 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0033/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 34. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 1, [[1/5, 4/5], [3/4, 1/4]], 4/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0034/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 35. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 1, [[2/5, 3/5], [3/4, 1/4]], 3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0035/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 36. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 1, [[3/5, 2/5], [3/4, 1/4]], 2/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0036/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 37. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=1. Give (from,to,steps,P^steps,its from-to entry).

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

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 1, [[4/5, 1/5], [3/4, 1/4]], 3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0037/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 38. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(0, 1, 2, [[6/25, 19/25], [19/80, 61/80]], 19/25)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 2, [[6/25, 19/25], [19/80, 61/80]], 19/25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0038/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 39. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(1, 0, 2, [[31/100, 69/100], [23/80, 57/80]], 23/80)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 2, [[31/100, 69/100], [23/80, 57/80]], 23/80).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0039/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 40. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(0, 1, 2, [[23/50, 27/50], [27/80, 53/80]], 27/50)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 2, [[23/50, 27/50], [27/80, 53/80]], 27/50).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0040/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 41. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(1, 0, 2, [[69/100, 31/100], [31/80, 49/80]], 31/80)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 2, [[69/100, 31/100], [31/80, 49/80]], 31/80).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0041/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 42. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(0, 1, 2, [[11/25, 14/25], [7/20, 13/20]], 14/25)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 2, [[11/25, 14/25], [7/20, 13/20]], 14/25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0042/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 43. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(1, 0, 2, [[23/50, 27/50], [9/20, 11/20]], 9/20)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 2, [[23/50, 27/50], [9/20, 11/20]], 9/20).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0043/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 44. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(0, 1, 2, [[14/25, 11/25], [11/20, 9/20]], 11/25)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 2, [[14/25, 11/25], [11/20, 9/20]], 11/25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0044/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 45. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(1, 0, 2, [[37/50, 13/50], [13/20, 7/20]], 13/20)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 2, [[37/50, 13/50], [13/20, 7/20]], 13/20).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0045/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 46. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(0, 1, 2, [[16/25, 9/25], [27/80, 53/80]], 9/25)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 2, [[16/25, 9/25], [27/80, 53/80]], 9/25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0046/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 47. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(1, 0, 2, [[61/100, 39/100], [39/80, 41/80]], 39/80)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 2, [[61/100, 39/100], [39/80, 41/80]], 39/80).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0047/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 48. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(0, 1, 2, [[33/50, 17/50], [51/80, 29/80]], 17/50)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 2, [[33/50, 17/50], [51/80, 29/80]], 17/50).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0048/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 49. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=1, to=0, steps=2. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(1, 0, 2, [[79/100, 21/100], [63/80, 17/80]], 63/80)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, 2, [[79/100, 21/100], [63/80, 17/80]], 63/80).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0049/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 50. A Conditional n-Step Entry without Inventing an Initial Law

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. from=0, to=1, steps=3. Give (from,to,steps,P^steps,its from-to entry).

View answer and explanation(0, 1, 3, [[119/500, 381/500], [381/1600, 1219/1600]], 381/500)

Keep row/column direction. At zero steps the entry is 1 exactly when the two state labels agree. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, 3, [[119/500, 381/500], [381/1600, 1219/1600]], 381/500).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0050/)

Finite-state Markov chain · qf-mod25-markovdeep-n-step-entry · candidate





## 51. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=0, to=1, steps=1. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

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

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 0, 1, 1, 1/4, 4/5, 1/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0051/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 52. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=1, to=0, steps=2. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/4, 3/4], 1, 0, 2, 3/4, 23/80, 69/320)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 1, 0, 2, 3/4, 23/80, 69/320).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0052/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 53. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=0, to=1, steps=3. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/4, 3/4], 0, 1, 3, 1/4, 589/1000, 589/4000)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 0, 1, 3, 1/4, 589/1000, 589/4000).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0053/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 54. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=1, to=0, steps=1. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/4, 3/4], 1, 0, 1, 3/4, 1/4, 3/16)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 1, 0, 1, 3/4, 1/4, 3/16).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0054/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 55. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=0, to=1, steps=2. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/4, 3/4], 0, 1, 2, 1/4, 14/25, 7/50)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 0, 1, 2, 1/4, 14/25, 7/50).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0055/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 56. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=1, to=0, steps=3. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/4, 3/4], 1, 0, 3, 3/4, 91/200, 273/800)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 1, 0, 3, 3/4, 91/200, 273/800).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0056/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 57. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=0, to=1, steps=1. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/4, 3/4], 0, 1, 1, 1/4, 2/5, 1/10)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 0, 1, 1, 1/4, 2/5, 1/10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0057/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 58. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=1, to=0, steps=2. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/4, 3/4], 1, 0, 2, 3/4, 13/20, 39/80)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 1, 0, 2, 3/4, 13/20, 39/80).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0058/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 59. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=0, to=1, steps=3. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/4, 3/4], 0, 1, 3, 1/4, 301/500, 301/2000)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 0, 1, 3, 1/4, 301/500, 301/2000).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0059/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 60. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=1, to=0, steps=1. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/4, 3/4], 1, 0, 1, 3/4, 3/4, 9/16)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 1, 0, 1, 3/4, 3/4, 9/16).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0060/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 61. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=0, to=1, steps=2. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/4, 3/4], 0, 1, 2, 1/4, 17/50, 17/200)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 0, 1, 2, 1/4, 17/50, 17/200).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0061/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 62. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/4, 3/4], from=1, to=0, steps=3. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/4, 3/4], 1, 0, 3, 3/4, 1263/1600, 3789/6400)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/4, 3/4], 1, 0, 3, 3/4, 1263/1600, 3789/6400).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0062/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 63. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=0, to=1, steps=1. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

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

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 0, 1, 1, 1/2, 4/5, 2/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0063/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 64. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=1, to=0, steps=2. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/2, 1/2], 1, 0, 2, 1/2, 23/80, 23/160)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 1, 0, 2, 1/2, 23/80, 23/160).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0064/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 65. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=0, to=1, steps=3. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/2, 1/2], 0, 1, 3, 1/2, 589/1000, 589/2000)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 0, 1, 3, 1/2, 589/1000, 589/2000).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0065/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 66. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=1, to=0, steps=1. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/2, 1/2], 1, 0, 1, 1/2, 1/4, 1/8)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 1, 0, 1, 1/2, 1/4, 1/8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0066/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 67. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=0, to=1, steps=2. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/2, 1/2], 0, 1, 2, 1/2, 14/25, 7/25)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 0, 1, 2, 1/2, 14/25, 7/25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0067/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 68. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=1, to=0, steps=3. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/2, 1/2], 1, 0, 3, 1/2, 91/200, 91/400)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 1, 0, 3, 1/2, 91/200, 91/400).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0068/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 69. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=0, to=1, steps=1. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

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

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 0, 1, 1, 1/2, 2/5, 1/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0069/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 70. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=1, to=0, steps=2. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/2, 1/2], 1, 0, 2, 1/2, 13/20, 13/40)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 1, 0, 2, 1/2, 13/20, 13/40).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0070/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 71. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=0, to=1, steps=3. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/2, 1/2], 0, 1, 3, 1/2, 301/500, 301/1000)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 0, 1, 3, 1/2, 301/500, 301/1000).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0071/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 72. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=1, to=0, steps=1. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/2, 1/2], 1, 0, 1, 1/2, 3/4, 3/8)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 1, 0, 1, 1/2, 3/4, 3/8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0072/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 73. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=0, to=1, steps=2. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/2, 1/2], 0, 1, 2, 1/2, 17/50, 17/100)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 0, 1, 2, 1/2, 17/50, 17/100).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0073/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 74. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=1, to=0, steps=3. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

View answer and explanation([1/2, 1/2], 1, 0, 3, 1/2, 1263/1600, 1263/3200)

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 1, 0, 3, 1/2, 1263/1600, 1263/3200).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0074/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 75. Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[3/4, 1/4], from=0, to=1, steps=1. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

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

This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3/4, 1/4], 0, 1, 1, 3/4, 4/5, 3/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0075/)

Finite-state Markov chain · qf-mod25-markovdeep-endpoint-joint-probability · candidate





## 76. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

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

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [4/5, 1/4], 1/5, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0076/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 77. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([1, 0, 1], [1/4, 3/5], 3/20, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/4, 3/5], 3/20, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0077/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 78. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([0, 1, 0], [2/5, 1/4], 1/10, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [2/5, 1/4], 1/10, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0078/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 79. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([1, 0, 1], [1/4, 1/5], 1/20, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/4, 1/5], 1/20, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0079/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 80. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

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

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [4/5, 1/2], 2/5, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0080/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 81. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

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

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/2, 3/5], 3/10, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0081/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 82. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

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

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [2/5, 1/2], 1/5, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0082/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 83. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

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

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/2, 1/5], 1/10, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0083/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 84. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

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

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [4/5, 3/4], 3/5, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0084/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 85. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([1, 0, 1], [3/4, 3/5], 9/20, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [3/4, 3/5], 9/20, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0085/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 86. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([0, 1, 0], [2/5, 3/4], 3/10, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [2/5, 3/4], 3/10, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0086/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 87. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([1, 0, 1], [3/4, 1/5], 3/20, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [3/4, 1/5], 3/20, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0087/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 88. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([0, 1, 0, 1], [4/5, 1/4, 4/5], 4/25, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 1], [4/5, 1/4, 4/5], 4/25, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0088/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 89. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([1, 0, 1, 0], [1/4, 3/5, 1/4], 3/80, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], [1/4, 3/5, 1/4], 3/80, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0089/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 90. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([0, 1, 0, 1], [2/5, 1/4, 2/5], 1/25, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 1], [2/5, 1/4, 2/5], 1/25, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0090/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 91. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([1, 0, 1, 0], [1/4, 1/5, 1/4], 1/80, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], [1/4, 1/5, 1/4], 1/80, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0091/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 92. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([0, 1, 0, 1], [4/5, 1/2, 4/5], 8/25, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 1], [4/5, 1/2, 4/5], 8/25, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0092/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 93. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([1, 0, 1, 0], [1/2, 3/5, 1/2], 3/20, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], [1/2, 3/5, 1/2], 3/20, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0093/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 94. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([0, 1, 0, 1], [2/5, 1/2, 2/5], 2/25, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 1], [2/5, 1/2, 2/5], 2/25, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0094/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 95. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([1, 0, 1, 0], [1/2, 1/5, 1/2], 1/20, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], [1/2, 1/5, 1/2], 1/20, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0095/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 96. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([0, 1, 0, 1], [4/5, 3/4, 4/5], 12/25, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 1], [4/5, 3/4, 4/5], 12/25, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0096/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 97. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([1, 0, 1, 0], [3/4, 3/5, 3/4], 27/80, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], [3/4, 3/5, 3/4], 27/80, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0097/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 98. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[3/5, 2/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0, 1] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([0, 1, 0, 1], [2/5, 3/4, 2/5], 3/25, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 1], [2/5, 3/4, 2/5], 3/25, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0098/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 99. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[4/5, 1/5], [3/4, 1/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[1, 0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([1, 0, 1, 0], [3/4, 1/5, 3/4], 9/80, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], [3/4, 1/5, 3/4], 9/80, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0099/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate





## 100. A Kernel-Started Path Excludes Any Initial-Law Mass

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

View answer and explanation([0, 1, 0, 1, 0], [4/5, 1/4, 4/5, 1/4], 1/25, conditional_on_initial_state)

Multiply only the transitions in their path order. This differs from an unconditional finite-path probability, which additionally requires an initial distribution. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0], [4/5, 1/4, 4/5, 1/4], 1/25, conditional_on_initial_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0100/)

Finite-state Markov chain · qf-mod25-markovdeep-conditional-path-probability · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261025-r06-finite-markov-chain-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261025-r06-finite-markov-chain-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261025-r06-finite-markov-chain-foundations/validation-report.json)