Question 11351: A Finite Path Includes Its Initial-State Mass

Original question and givens

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

• G001 — Use one fixed finite time-homogeneous transition table=[[1/3, 2/3], [1/2, 1/2]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require initial law=[1/2, 1/2] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.

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=[[1, 3], [2, 3]]

transition row0:normalized=[1/3, 2/3]

transition row0 total := sum(1/3, 2/3)

transition row0 total = 1

transition row0:weights=[1/3, 2/3];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

transition row1:raw rational pairs=[[1, 2], [1, 2]]

transition row1:normalized=[1/2, 1/2]

transition row1 total := sum(1/2, 1/2)

transition row1 total = 1

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

nonnegative=Yes;total_one=Yes;qualified=Yes

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

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

initial law:raw rational pairs=[[1, 2], [1, 2]]

initial law:normalized=[1/2, 1/2]

initial law total := sum(1/2, 1/2)

initial law total = 1

initial law:weights=[1/2, 1/2];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

path=[0, 1, 0];initial law=[1/2, 1/2]

factor0=initial[0]=1/2

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

factor1=P[0][1]=2/3

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

factor2=P[1][0]=1/2

path prefix product1 := mul(1/2, 2/3)

path prefix product1 = 1/3

path prefix product2 := mul(1/3, 1/2)

path prefix product2 = 1/6

ordered path factors=[1/2, 2/3, 1/2]

finite path probability=1/6

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], [1/2, 2/3, 1/2], 1/6)

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.