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).

p <- 0.4
q <- 0.6
r <- q/p


for (i in 0:8){
P <- round((1-r^i)/(1-r^8),4)
print (P)
}
## [1] 0
## [1] 0.0203
## [1] 0.0508
## [1] 0.0964
## [1] 0.1649
## [1] 0.2677
## [1] 0.4219
## [1] 0.6531
## [1] 1

The probability that he wins 8 dollars before losing all of his money if (a) he bets 1 dollar each time (timid strategy) is 0.0203.

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

The quickest way to for him to get up to 8 dollars is to win 3 times doubling each time (1,2,4) in a row. With a probability of winning 0.4 against the guard we can derive the following equation:

\(\ P=(0.4)^3 = 0.064\)

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

Comparing the probabilities of each strategy, the bold strategy of 0.064 gives him a better and higher chance than the timid strategy of 0.0203.