Chapter 9.8, Problem 23ES

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

Chapter
Section

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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A gambler repeatedly bets that a die will come up 6 when rolled. Each time the die comes up 6, the gambler wins $1; each time it does not, the gambler loses$1. He will quit playing either when he is ruined or when he wins \$300. If P n is the probability that the gambler is ruined when he begins play with S n , then P k − 1 = 1 6 P k + 5 6 P k − 2 for every integer k with 2 ≤ k ≤ 300. Also P 0 = 1 and P 300 = 0. Find an explicit formula for P n and use it to calculate P 20 . (Exercise 33 in Section 9.9 asks you to derive the recurrence relation for this sequence.)

To determine

To find an explicit formula for Pn and use it to calculate P20.

Explanation

Given information:

The given recurrence relation is

Pk1=16Pk+56Pk2, for every integer with 2k300P0=1P300=0

Calculation:

Pk1=16Pk+56Pk26Pk1=Pk+5Pk2{on multiplying both sides by 6}Pk6Pk1+5Pk2=0r26r+5=0{replacing Pk=r2 and thus Pk2=r0}

On solving the above characteristic equation the solutions are r=1,5

The general solution of the recurrence relation is of the form

Pn=C1(r1)n+C2(r2)n{where r1 and r2 are the distinct roots of the characteristic equation and C1 and C2are the arbitary constants}Pn=C1(1)n+C2(5)n{r1=1r2=5}

Now let us apply the initial conditions

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