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
(a) he bets 1 dollar each time (timid strategy).
q <- 0.6
p <- 0.4
s <- 1
M <- 8
qp <- q/p
qp
## [1] 1.5
(qp^1-1)/(qp^8-1)
## [1] 0.02030135
There is a 2% chance here.
(b) he bets, each time, as much as possible but not more than necessary tobring his fortune up to 8 dollars (bold strategy).
(0.4) ^ 3
## [1] 0.064
OR
dbinom(3,3,0.4)
## [1] 0.064
There is a 6.4% chance with bold strategy.
(c) Which strategy gives Smith the better chance of getting out of jail?
The bold strategy gives him better chance i.e 6.4%