Question 11394: Two-Step Probability by Summing Every Intermediate State

Original question and givens

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

• G001 — Use one fixed finite time-homogeneous transition table=[[19/21, 2/21], [10/11, 1/11]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.

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=[[19, 21], [2, 21]]

transition row0:normalized=[19/21, 2/21]

transition row0 total := sum(19/21, 2/21)

transition row0 total = 1

transition row0:weights=[19/21, 2/21];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

transition row1:raw rational pairs=[[10, 11], [1, 11]]

transition row1:normalized=[10/11, 1/11]

transition row1 total := sum(10/11, 1/11)

transition row1 total = 1

transition row1:weights=[10/11, 1/11];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

complete transition matrix=[[19/21, 2/21], [10/11, 1/11]];row sums=[1, 1]

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

from=0;to=1;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(19/21, 2/21)

two-step middle0 term = 38/441

two-step middle1 term := mul(2/21, 1/11)

two-step middle1 term = 2/231

two-step probability := sum(38/441, 2/231)

two-step probability = 460/4851

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

(0, 1, [38/441, 2/231], 460/4851)

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.