Hazal Gunduz
Exercise: 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) (timid strategy)
p <- 0.4
q <- 0.6
r <- q / p
A <- (1 - (q / p)^1) / (1 - (q / p)^8)
A## [1] 0.02030135The probability that he wins 8 dollars before losing all of his money if (a) he bets 1 dollar each time (timid strategy) is 0.0203
(b) (bold strategy)
Since Smith tries to double his money each time, he can’t reach every integer before 8. It can go from 1 to 0 or 2. It can go from 2 to 4 or 0. Finally it can go from 4 to 8 or 0.
B <- dbinom(3, 3, p)
B## [1] 0.064The probability that he wins 8 dollars before losing all of his money if (b) he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy) is 0.064
(c) (better strategy)
Comparing the probabilities of strategies, the bold strategy of “0.064” gives to Smith a better and higher chance than the timid strategy of “0.0203”.