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
Based on The Gambler’s Ruin Problem:
If \(p \neq q\) \[P_{i} = \frac{1-(q/p)^{t}}{1-(q/p)^{N}}\]
(a) he bets 1 dollar each time (timid strategy).
p <- 0.4
q <- 1 - p
i <- 1
N <- 8
round((1-(q/p)**i)/(1-(q/p)**N), 4)## [1] 0.0203(b) he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).
Probability with the assumption he wins each bet, after 1st win gets $2, then 2nd win gets $4 and third win gets $8
p**3## [1] 0.064(c) Which strategy gives Smith the better chance of getting out of jail?
Smith has a better chance of getting out of jail using the bold strategy.