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
- he bets 1 dollar each time (timid strategy)
- he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).
- which strategy gives Smith the better chance of getting out of jail?
# (a)
p = 0.4
q = 0.6
prob_timid = (1-q/p)/(1-(q/p)^8)
prob_timid## [1] 0.02030135# (b) he bets all his money to win the same amount or lose all. So he has to win in a row, otherwise, he lose everthing.
prob_bold = p^3
prob_bold## [1] 0.064- prob_timid is smaller than prob_bold. I think bold strategy will be better.