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).
p <- 0.4
q <- 0.6

(1 - (q/p)^1) / ( 1 - (q/p)^8)
## [1] 0.02030135
  1. he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).

Bet 1 Won: $2, Wager: $2 Bet 2 Won: $4, Wager: $4 Bet 3 Won: $8, Wager: $0

In order for Smith to raise money using this approach, he would need to bet 3 times and win all. Each win has the same probability 0.4. The probability that Smith can raise the money needed for bail using this approach would be p ^ # of bets, p^3.

p^3
## [1] 0.064
  1. Which strategy gives Smith the better chance of getting out of jail?

The second staratgey, gives a better chance of getting out of jail.