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?

Using Gambler’s Ruin formula:

\[p = \frac{1 - (p/q)^s}{1 - (q/p)^M}\]

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

## variables
P <- 0.4
q <- 0.6
M <- 8
s <- 1
(P_1_dollar <- (1 - (q/P)^s) / (1 - (q/P)^M))
## [1] 0.02030135

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

##  variables
x <- 3
s <- 3
p <- 0.4

dbinom(x, s, p)
## [1] 0.064

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

The bold strategy gives Smith the better chance of getting out of jail 6.4%