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

\(q = 0.6\) \(p = 0.4\) \(s = 1\) \(M = 8\) \(\frac{q}{p} = \frac{0.6}{0.4} = 1.5\)

\(P = \frac{1-(\frac{q}{p})^s}{1-(\frac{q}{p})^M} \\ P = \frac{1-1.5^{1}}{1-1.5^{8}}\)

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

Solution

(1.5^1-1)/(1.5^8-1)
## [1] 0.02030135

(b) he bets, each time, as much as possible but not more than necessary to

bring his fortune up to 8 dollars (bold strategy).

Solution

dbinom(3,3,0.4)
## [1] 0.064

Check

markov <- matrix(c(1,0,0,0,0,0.6,0,0.4,0,0,0.6,0,0,0.4,0,0.6,0,0,0,0.4,0,0,0,0,1), ncol=5,nrow=5, byrow = TRUE)
markov
##      [,1] [,2] [,3] [,4] [,5]
## [1,]  1.0    0  0.0  0.0  0.0
## [2,]  0.6    0  0.4  0.0  0.0
## [3,]  0.6    0  0.0  0.4  0.0
## [4,]  0.6    0  0.0  0.0  0.4
## [5,]  0.0    0  0.0  0.0  1.0
pi <- matrix(c(0,1,0,0,0), ncol=5,nrow = 1,byrow = TRUE)
pi_1 <- pi%*%markov
pi_2 <- pi_1%*%markov
pi_3 <- pi_2%*%markov
pi_4 <- pi_3%*%markov
pi_4
##       [,1] [,2] [,3] [,4]  [,5]
## [1,] 0.936    0    0    0 0.064

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

Solution

The “Bold Strategy” triples his probability of winning. Whereas, the “Timid Strategy” does give him the ability to lose bets.