DATA 605 FUNDAMENTALS OF COMPUTATIONAL MATHEMATICS

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
z <- 1
M <- 8

(q_z <- (((q/p)^z - 1)/((q/p)^M - 1)))

## [1] 0.02030135

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

p <- 0.4
M <- 8
wins <- 0
trials <- 1e+05

for (i in 1:trials) {
    z <- 1
    while (z > 0 & z < M) {
        if (runif(1) <= p) {
            z <- z + z
        } else {
            z <- z - z
        }
    }
    if (z >= M) {
        wins <- wins + 1
    }
}

wins/trials

## [1] 0.06443

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

He should go bold!