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:
He bets 1 dollar each time (timid strategy).
He bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).
Which strategy gives Smith the better chance of getting out of jail?
Define the universal variables used in the game.
# Define the probability of losing (q).
q <- 0.6
# Define the probability of winning (p).
p <- 0.4
# Calculate the probability ratio of winning to losing (q/p).
ratio <- q/p
# Define the minimum bet amount (x), and the amount needed to win the bet (y).
x <- 1
y <- 8
(a) Find the probability that he wins 8 dollars before losing all of his money if he bets 1 dollar each time (timid strategy).
We can calculate the probability of the prisoner winning $8 before losing all of his money utilizing the following formula: \[P_{x} = \frac{(1 - \frac{q}{p})^{x}}{(1 - \frac{q}{p})^{y}}\]
Using the above formula, we can use R to calculate the probability of the prisoner winning as follows:
# Calculate the probability of the prisoner winning the game (timid strategy).
win_probability <- (1 - ratio^x) / (1 - ratio^y)
probability_prisoner_wins <- round(win_probability, 2)
probability_prisoner_wins## [1] 0.02Answer: The probability that the prisoner will win the game is 0.02.
(b) He bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).
In this scenario, we know that the prisoner will bet all of his money each time he makes a bet. Therefore, the betting pattern will look like this:
First bet: The prisoner only has $1 at this stage so his maximum bet will be $1. If he wins, he will have $2 for the next bet.
Second bet: Assuming the prisoner won the previous bet, he will bet $2 for this bet. If he wins, he now has $4 for the next bet.
Final bet: If the prisoner makes it to this stage, he will bet $4. If he wins, he will have $8, which means he has won the game and the game ends.
According to the above betting pattern, we know that the prisoner must win 3 consecutive bets in order to win the game and make bail. The probability of him winning the game is 0.4, therefore we can calculate the probability of him winning the $8 using \(p^3\).
# Calculate the probability of the prisoner winning the game (bold strategy).
probability_prisoner_wins <- round(p^3, 2)
probability_prisoner_wins## [1] 0.06Answer: The probability that the prisoner will win the $8 and make bail is 0.06.
(c) Which strategy gives Smith the better chance of getting out of jail?
Answer: The bold strategy gives Smith a better chance of getting out of jail. Using the timid strategy, he has a 0.2 chance of winning whereas the bold strategy gives him a 0.06 chance of winning.