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

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

x<-1 -((1.5- (1.5^8))/(1- (1.5^8)))
x

## [1] 0.02030135

Probability is .020.

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

state<-matrix(c(1,0.6,0.6,0.6,0,0,0,0,0,0,0,0.4,0,0,0,0,0,0.4,0,0,0,0,0,0.4,1),nrow = 5, byrow=FALSE)
initial<-matrix(c(0,1,0,0,0),nrow=1)
x1=initial%*%state
x2=x1%*%state
x3=x2%*%state
x3

##       [,1] [,2] [,3] [,4]  [,5]
## [1,] 0.936    0    0    0 0.064

Probability is .064

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

Since the bold strategy has a higher probability just on that observation alone, I would say the bold strategy would give Smith the better chance to get out of jail.