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). Pwin = 1−((0.6/0.4)1−(0.6/0.4)8)/11(0.6/0.4)8Pwin = 1−((0.6/0.4)1−(0.6/0.4)8)/11(0.6/0.4)8 = 0.02030135

  2. he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).

dbinom(3,3,0.4)
## [1] 0.064
  1. Which strategy gives Smith the better chance of getting out of jail?

As we see from a and b, the bold strategy has a probability of getting out of jail as 0.06, which is more than the probability while using the timid strategy, which is 0.02. Hence bold strategy has a better chance for him getting out of jail.