Question

Consider again the three different conditions (a), (b), and (c) given in Exercise 2, but suppose now 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

a. Condition (a) has greatest probability

b. Condition (b) has probability more than condition (c) and less than (a)

c. Condition (c) has lowest probability

Step-by-step solution

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

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

Compute the probability for (a)

a.

Assuming that it is an unfair 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.

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

Here \(0 < p < \frac{1}{2}\)

Let \(\left( {\frac{{1 - p}}{p}} \right) = r\)

Here r is always greater than 1 since p is less than 0.5

Then \({a_i} = \frac{{{r^i} - 1}}{{{r^k} - 1}}\) where \(i = 1,2,...,k - 1\)

\(\begin{aligned}{l}\therefore for\,i = 2\,and\,k = 3\,\\{a_2} = \frac{{{r^2} - 1}}{{{r^3} - 1}}\end{aligned}\)

Compute the probability for (b)

b.

Proceeding same as above

\(\begin{aligned}{l}for\,i = 20\,\,and\,k = 30\\{a_{20}} = \frac{{{r^{20}} - 1}}{{{r^{30}} - 1}}\end{aligned}\)

Compute the probability for (c)  

c.

Proceeding similarly as above

\(\begin{aligned}{l}for\,i = 200\,\,and\,k = 300\\{a_{200}} = \frac{{{r^{200}} - 1}}{{{r^{300}} - 1}}\end{aligned}\)

Comparing the probability for (a), (b) and (c)  

Clearly seeing the ratio, we can state that

\({a_2} > {a_{20}} > {a_{200}}\)