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?

Solution:

  1. Smith has to win 1st game, the last two, and at least 4 in between. \(0.4^7+0.4^8*0.6*\binom{5}{1}+...=0.0924\)
  2. \(0.4^3=0.064\)
  3. a
mypr<-0.4^7

for (i in (1:1000000)){
  mypr<-mypr+0.4^3*dbinom(i,size=3+i*2,0.6)
}

mypr
## [1] 0.09235721
mypr1<-0.4^3

mypr1
## [1] 0.064