Homework 10, part A

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

  1. he bets 1 dollar each time (timid strategy).

This is the “The Gambler’s Ruin Problem” (p. 487, probability book)

P(w) = 0.4

P(l) = 0.6

if l is not equal to w, then P(winning) = ((l/w)^1 - 1)/((l/w)^8-1)

w <- 0.4
l <- 0.6
z = 1
m = 8

((l/w)^z - 1)/((l/w)^m-1)
## [1] 0.02030135

0.02030135 is the probability that Smith wins 8 dollars before losing all of his money if he bets 1 dollar each time.

  1. if he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).

If he does win his purse will follow the following sequence: 1,2,4,8. If he starts with 1 dollar and must win 3 bets in a row

P(w) = 0.4

dbinom(3,3,0.4) 
## [1] 0.064

0.064 is the probability that he wins 8 dollars before losing all of his money (bold strategy)

  1. Which strategy gives Smith the better chance of getting out of jail?

Bold strategy gives Smith the better chance of getting out of jail (more than triples).