Question 11371: A Finite Path Includes Its Initial-State Mass

Original question and givens

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

• G001 — Use one fixed finite time-homogeneous transition table=[[21/23, 2/23], [11/12, 1/12]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[21/22, 1/22] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

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=[[21, 23], [2, 23]]

transition row0:normalized=[21/23, 2/23]

transition row0 total := sum(21/23, 2/23)

transition row0 total = 1

transition row0:weights=[21/23, 2/23];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

transition row1:raw rational pairs=[[11, 12], [1, 12]]

transition row1:normalized=[11/12, 1/12]

transition row1 total := sum(11/12, 1/12)

transition row1 total = 1

transition row1:weights=[11/12, 1/12];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

complete transition matrix=[[21/23, 2/23], [11/12, 1/12]];row sums=[1, 1]

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

initial law:raw rational pairs=[[21, 22], [1, 22]]

initial law:normalized=[21/22, 1/22]

initial law total := sum(21/22, 1/22)

initial law total = 1

initial law:weights=[21/22, 1/22];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

path=[0, 1, 0];initial law=[21/22, 1/22]

factor0=initial[0]=21/22

path step=1;current=0;next=1

factor1=P[0][1]=2/23

path step=2;current=1;next=0

factor2=P[1][0]=11/12

path prefix product1 := mul(21/22, 2/23)

path prefix product1 = 21/253

path prefix product2 := mul(21/253, 11/12)

path prefix product2 = 7/92

ordered path factors=[21/22, 2/23, 11/12]

finite path probability=7/92

Reverse checks

• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Recompute path prefix product2 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute path prefix product1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute initial law total 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, 0], [21/22, 2/23, 11/12], 7/92)

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.