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Chapter 2: Q2E (page 89)

Consider the following three different possible conditions in the gambler’s ruin problem:

a. The initial fortune of gambler A is two dollars, and the initial fortune of gambler B is one dollar.

b. The initial fortune of gambler A is 20 dollars, and the initial fortune of gambler B is 10 dollars.

c. The initial fortune of gambler A is 200 dollars, and the initial fortune of gambler B is 100 dollars.

Suppose that p = 1/2. For which of these three conditions is there the greatest probability that gambler A will win the initial fortune of gambler B before he loses his own initial fortune?

Short Answer

Expert verified

The probability that gambler A will win the initial fortune of gambler B before he loses his own initial fortune in 3 different cases is

a. \({a_i} = \frac{2}{3}\)

b. \({a_i} = \frac{2}{3}\)

c. \({a_i} = \frac{2}{3}\)

Step by step solution

01

Given information

Initial fortune of A\(i = 2,20,200\)

Initial fortune of B\(k - i = 1,10,100\)

Therefore, the total fortune of the 2 gamblers\(k = 3,30,300\)

the probability that gambler A will win one dollar from gambler B is p and the probability that gambler B will win one dollar from gambler A is 1 – p

\(p = \frac{1}{2}\)

02

compute the probability for case (a)

a.

Assuming that it is a fair play

Required probability is given by\({a_i}\)

Let \({a_i}\) denote the probability that the fortune of gambler A will reach k dollars before it reaches 0 dollars, given that his initial fortune is i dollars.

\({a_i} = \frac{i}{k};i = 1,2,...,k - 1\)

03

compute the probability for case (b)

b.

Assuming that it is a fair play

Required probability is given by\({a_i}\)

Let \({a_i}\) denote the probability that the fortune of gambler A will reach k dollars before it reaches 0 dollars, given that his initial fortune is i dollars.

\({a_i} = \frac{i}{k};i = 1,2,...,k - 1\)

04

compute the probability for case (c)

c.

Assuming that it is a fair play

Required probability is given by\({a_i}\)

Let \({a_i}\) denote the probability that the fortune of gambler A will reach k dollars before it reaches 0 dollars, given that his initial fortune is i dollars.

\({a_i} = \frac{i}{k};i = 1,2,...,k - 1\)

05

conclusion

These three conditions have the same probability that gambler A will win the initial fortune of gambler B before he loses his own initial fortune.

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Most popular questions from this chapter

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