Homework10

10: 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

(a) he bets 1 dollar each time (timid strategy).

# Define the parameters
p <- 0.4
q <- 0.6
z <- 1  # Number of dollars needed to reach 8 dollars
m <- 8  # Total amount of money he has

# Calculate the probability using the formula
prob_a <- (((q/p)^z)-1) /( ((q/p)^m) - 1)
print(paste("Probability of winning 8 dollars before losing all money (strategy a):", prob_a))

## [1] "Probability of winning 8 dollars before losing all money (strategy a): 0.0203013481363997"

2.03%

(b) 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

prob_bold <- p^3
print(paste("Probability of winning 8 dollars before losing all money (bold strategy):", prob_bold))

## [1] "Probability of winning 8 dollars before losing all money (bold strategy): 0.064"

6.4%

(c) Which strategy gives Smith the better chance of getting out of jail?

The probability for the timid strategy (part a) is 2.03%.

The probability for the bold strategy (part b) is 6.4%.

Since the probability of winning 8 dollars before losing all money is higher for the bold strategy, it is the better strategy in this scenario. Based on the calculated probabilities, the bold strategy is better for Smith to get out of jail.