Question 11373: A Finite Path Includes Its Initial-State Mass

Original question and givens

Use one fixed finite time-homogeneous transition table=[[23/25, 2/25], [12/13, 1/13]]. 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=[23/24, 1/24] 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=[[23/25, 2/25], [12/13, 1/13]]. 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=[23/24, 1/24] 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=[[23, 25], [2, 25]]

transition row0:normalized=[23/25, 2/25]

transition row0 total := sum(23/25, 2/25)

transition row0 total = 1

transition row0:weights=[23/25, 2/25];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

transition row1:raw rational pairs=[[12, 13], [1, 13]]

transition row1:normalized=[12/13, 1/13]

transition row1 total := sum(12/13, 1/13)

transition row1 total = 1

transition row1:weights=[12/13, 1/13];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

complete transition matrix=[[23/25, 2/25], [12/13, 1/13]];row sums=[1, 1]

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

initial law:raw rational pairs=[[23, 24], [1, 24]]

initial law:normalized=[23/24, 1/24]

initial law total := sum(23/24, 1/24)

initial law total = 1

initial law:weights=[23/24, 1/24];sum=1

nonnegative=Yes;total_one=Yes;qualified=Yes

path=[0, 1, 0];initial law=[23/24, 1/24]

factor0=initial[0]=23/24

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

factor1=P[0][1]=2/25

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

factor2=P[1][0]=12/13

path prefix product1 := mul(23/24, 2/25)

path prefix product1 = 23/300

path prefix product2 := mul(23/300, 12/13)

path prefix product2 = 23/325

ordered path factors=[23/24, 2/25, 12/13]

finite path probability=23/325

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], [23/24, 2/25, 12/13], 23/325)

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.