Question 11381: Two-Step Probability by Summing Every Intermediate State

Original question and givens

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

• G001 — Use one fixed finite time-homogeneous transition table=[[3/4, 1/4], [7/9, 2/9]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-finite-markov-chain

Step-by-step derivation

transition row0:raw rational pairs=[[3, 4], [1, 4]]

transition row0:normalized=[3/4, 1/4]

transition row0 total := sum(3/4, 1/4)

transition row0 total = 1

transition row0:weights=[3/4, 1/4];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

transition row1:raw rational pairs=[[7, 9], [2, 9]]

transition row1:normalized=[7/9, 2/9]

transition row1 total := sum(7/9, 2/9)

transition row1 total = 1

transition row1:weights=[7/9, 2/9];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

complete transition matrix=[[3/4, 1/4], [7/9, 2/9]];row sums=[1, 1]

nonnegative=Yes;row_stochastic=Yes;rows=current state, columns=next state

from=1;to=0;middle states=[0, 1]

two-step conditional entry=sum_k P[from][k]*P[k][to];no initial law needed

two-step middle0 term := mul(7/9, 3/4)

two-step middle0 term = 7/12

two-step middle1 term := mul(2/9, 7/9)

two-step middle1 term = 14/81

two-step probability := sum(7/12, 14/81)

two-step probability = 245/324

Reverse checks

• S011 · premise_and_result_audit — Recompute two-step probability with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute two-step middle1 term with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute two-step middle0 term with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute transition row1 total with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute transition row0 total with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(1, 0, [7/12, 14/81], 245/324)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.