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).

Answer:

\(P = \frac{1 - \frac{q}{p}^j}{1 - \frac{q}{p}^N}\)

p <- 0.4
q <- 1-p
j <- 1
N <- 8
P <- (1 - (q/p)^j) / ( 1 - (q/p)^N)
P
## [1] 0.02030135
  1. he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).

Answer: Smith must win up to 8 dollar. He needs to win successive 3 times.

P_bold <- p^3
P_bold
## [1] 0.064
  1. Which strategy gives Smith the better chance of getting out of jail?

Answer: The bold strategy has a probability of 0.064 higher than timid strategy probability 0.02030135. Bold strategy gives Smith the better chance of getting out of jail.