Explanation Given: The casino manager tells one person about losing the slots. The probability that the person would tell the truth is p . The probability of the decision being passed on is p . Explanation: The probailities for the correct decision being passed on can be represented by a transition matrix of order 2 × 2 . The first row would give the probabilites for the person passing the correct decision and it being heard by the people or not. The second row would correspond to the person passing the wrong decision. Use the provided information to set up the transition matrix as P . P = [ p 1 − p 1 − p p ] Clearly the sum of all entries for each row is 1. This also holds for each column. The long-range probabilties for the fraction of people that will hear the correct decision can be computed as the equilibrium vector for the transition matrix P . A vector v is said to be an equilibrium vector for a transition matrix P , if the following condition holds: v P = v Let the vector v be defined as follows, so that the multiplication holds. [ v 1 v 2 ] Then, v P = v [ v 1 v 2 ] [ p 1 − p 1 − p p ] = [ v 1 v 2 ] Simplify the left-hand side to obtain, [ p v 1 + ( 1 − p ) v 2 ( 1 − p ) v 1 + p v 2 ] = [ v 1 v 2 ] This gives rise to the following system of equations, p v 1 + ( 1 − p ) v 2 = v 1 ( 1 − p ) v 1 + p v 2 = v 2

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ISBN: 9780321979438

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 10.2, Problem 38E

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The long-range probabilty for the fraction of people that will hear the correct decision.

Solution Summary: The author explains how the long-range probabilities for the tion of people that will hear the correct decision can be computed as the equilibrium vector for a transition matrix P.

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