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Chapter 2: Q2E (page 89)

Consider the following three different possible conditions in the gambler鈥檚 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 鈥� 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

Suppose that a box contains one blue card and four red cards, which are labelled A, B, C, and D. Suppose also that two of these five cards are selected at random, without replacement.

a. If it is known that card A has been selected, what is the probability that both cards are red?

b. If it is known that at least one red card has been selected, what is the probability that both cards are red?

Suppose a person rolls two balanced dice twice in succession. Determine the probability that the sum of the two numbers that appear on each of the three rolls will be 7.

Suppose that two players A and B take turns rolling a pair of balanced dice and that the winner is the first player who obtains the sum of 7 on a given roll of the two dice. If A rolls first, what is the probability that B will win?

Suppose that a certain boxAcontains five balls and another boxBcontains 10 balls. One of these two boxes isselected at random and one ball from the selected box is transferred to the other box. If this process of selecting abox at random and transferring one ball from that box to the other box is repeated indefinitely, what is the probability that boxAwill become empty before boxBbecomes empty?

Consider again the box containing the five different coins described in Exercise 7. Suppose that one coin is selected at random from the box and is tossed repeatedly until a head is obtained.

a. If the first head is obtained on the fourth toss, what is the posterior probability that theith coin was selected (i=1, . . .,5)?

b. If we continue to toss the same coin until another head is obtained, what is the probability that exactly three additional tosses will be required?
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