Question

Smith is in jail and has 1 dollar; he can get out on bail if he has 8 dollars. A guard agrees to make a series of bets with him. If Smith bets A dollars, he wins A dollars with probability .4 and loses A dollars with probability .6. Find the probability that he wins 8 dollars before losing all of his money if

(a)he bets 1 dollar each time (timid strategy).

(b)he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).

(c)Which strategy gives Smith the better chance of getting out of jail?

Solution (a) - Timid Strategy

Using Gambler ruin Probability method

\(P_{z} = \frac{1-(q/p)^{z}}{1 - (q/p)^{m}}\) \(where, q \neq p , 0 \leq z \leq m\)

and \(P_{z} =\) the probability that the gambler’s stake reaches m without ever having reached 0.

z = 1
m = 8
p = 0.4
q = 0.6

prob_timid = round(((1-(q/p)^z) / (1-(q/p)^m)), 4)
prob_timid
## [1] 0.0203

Using the Timid Strategy, the probability that he wins 8 dollars before losing all of his money if he bets 1 dollar each time is 0.0203


Solution (b) - Bold Strategy

If he uses the bold strategy, it means that he would bet all the money each time he wins to get his fortune up to 8 dollars as soon as possible. So, in this case there are only 4 steps (3 trials): 1, 2 (1st win), 4 (2nd win), 8 (3rd win) to reach $8.
1st trial: Bet 1 dollars to get another 1 dollar and get fortune to 2 dollars.
2nd trial: Bet all 2 dollars to get another 2 dollars to bring fortune up to 4 dollars.
3rd trial: Bet all 4 dollars to get another 4 dollars to bring fortune up to 8 dollars.
NB: For each trial, he has the same probability of success (p).

Because this is bold strategy, we can also solve this problem by using the Binomial distribution.
Binomial method:

Number of trials, n = 3

Number of successes, x = 3

\(_{x}^{n}\textrm{C}p^{x}(1-p)^{n-x}\)

\(_{3}^{3}\textrm{C}*0.4^{3}*(1-0.4)^{3-3}\)

= 0.064

prob_bold = round(dbinom(3, 3, 0.4), 4)
prob_bold
## [1] 0.064

Using the Bold Strategy, the probability that he wins 8 dollars before losing all of his money if he bets all each time is 0.064


Solution (c)

The Bold strategy gives Smith approximately 3 times better chance of getting out of jail than the Timid strategy. hence, I will recommend the Bold Strategy.