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 <- 1-p
j <- 1
N <- 8

P <- (1-(q/p)^j) / (1-(q/p)^N)
P
## [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).
r <- 10000
w <- 0 

for (i in 1:r){
  j <- 1
  while (j > 0 & j < N){
    if(runif(1) <= p){
      j <- j + j
    }else{
      j <- j-j
    }
  }
  if(j >= N){
    w <- w + 1
  }
}

w/r
## [1] 0.0616
  1. Which strategy gives Smith the better chance of getting out of jail?

The Bold strategy will give smith a better chance to get out of jail.