91Ó°ÊÓ

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

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

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 \(i = 1,2,...,k - 1\)

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

b.

Proceeding same as above

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

c.

Proceeding similarly as above

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

Clearly seeing the ratio, we can state that

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