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:

*The Gambler’s Ruin Problem: \[ \begin{aligned} &P_{i}=\frac{1-\left(\frac{q}{p}\right)^{t}}{1-\left(\frac{q}{p}\right)^{N}}, \text { if } p \neq q \\ &P_{i}=\frac{i}{N}, \text { if } p=q=0.5 \end{aligned} \]

he bets 1 dollar each time (timid strategy).

p <- 0.4
q <- 1-p
i <- 1
N <- 8
round(((1-(q/p)**i))/(1-(q/p)**N),5)
## [1] 0.0203

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

Assuming he wins each bet he will get: 1, 2 (1st win), 4 (2nd win), 8 (3rd win)

p**3
## [1] 0.064

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

Smith is better off taking the bold approach as he has a 6.4% chance of winning the 8 dollars he needs to post bail