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

Answer

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

\[ P = \frac {1 - (\frac {q}{p}) ^ s } {1 - (\frac {q}{p}) ^ M} \]

Given, q = 0.6
p = 0.4,
s = 1,
M = 8

According to formula of Gambler’s Ruin:
\[ P = \frac {1 - (\frac {q}{p}) ^ s } {1 - (\frac {q}{p}) ^ M} = \frac {1 - (\frac {0.6}{0.4}) ^ 1 } {1 - (\frac {0.6}{0.4}) ^ 8} = 0.02\]

# using R
q <- 0.6
p <- 0.4
s <- 1
M <- 8
P <- (1 -  (q/p) ^ s) / (1 - (q/p) ^ M)
print(P)
## [1] 0.02030135

The probability of wining is 0.02.

  1. 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 bets his entire money each time until 8 dollar. He starts with 1 dollar and must win 3 bets in a row with probability of 0.4

At first bet, If he wins, he ends up with $2.

At second bet, If he wins, he ends up with $4.

At third bet, If he wins, he ends up with $8. 

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

The probability of wining is 0.064.

  1. The bold strategy is having higher probability than timid strategy, So, bold strategy has a better chance of getting out of jail.