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). (b) he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy). (c) Which strategy gives Smith the better chance of getting out of jail?

k <- 1
M <- 8
p <- 0.4
q <- 0.6
r <- q/p

A) he bets $1 each time (timid strategy)

(1 - (q/p)^k)/ (1-(q/p)^M)
## [1] 0.02030135

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

dbinom(3,3,p)
## [1] 0.064

C) Which is better? Bold!

When doing this before I chose the timid strategy but now doing it with R code I can say that the bold strategy is the better choice to go