(a) Explanation Given Information: As variation of the gambler’s run problem, suppose that on each play, the probability of winning is 1 2 and the gambler stops playing if he accumulates $ 4 or goes broke. Formula Used: The stable matrix of A is: [ I S ( I − R ) − 1 0 0 ] Note: The identity matrix I in ( I − R ) − 1 is chosen to be the same sizes as R in order to make the matrix subtraction permissible. Calculation: Consider the matrix, $ 0 $ 4 $ 1 $ 2 $ 3 $ 0 $ 4 $ 1 $ 2 $ 3 [ 1 0 1 2 0 0 0 1 0 0 1 2 0 0 0 1 2 0 0 0 1 2 0 1 2 0 0 0 1 2 0 ] From above matrix the R = [ 0 1 2 0 1 2 0 1 2 0 1 2 0 ] , S = [ 1 2 0 0 0 0 1 2 ] . Now calculate ( I − R ) , ( I − R ) = [ 1 0 0 0 1 0 0 0 1 ] − [ 0 1 2 0 1 2 0 1 2 0 1 2 0 ] = [ 1 − 1 2 0 − 1 2 1 − 1 2 0 − 1 2 1 ] Now calculate the value of F = ( I − R ) (b) To determine To calculate: The expected number of times that he will play before quitting.

Finite Mathematics & Its Applications (12th Edition)

12th Edition

ISBN: 9780134437767

Author: Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair

Publisher: PEARSON

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Chapter 8.3, Problem 29E

(a)

To determine

To calculate: The probability of his eventually going broke if he start with $1,$2 and $3.

(b)

To determine

To calculate: The expected number of times that he will play before quitting.

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Chapter 8.3, Problem 28E

Chapter 8.3, Problem 30E

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Finite Mathematics & Its Applications (12th Edition)

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The probability of his eventually going broke if he start with $ 1 , $ 2 and $3 .

Solution Summary: The author calculates the probability of a gambler going broke if he starts with 1,2and 3. The identity matrix I is chosen to be the same sizes as R to