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
Also known as gamblers ruin:
We are given:
Initial stake \(z = k = 1\).
\(M = 8\)
\(P = 0.4\)
\(q = 0.6\)
\(q_z = \frac{(\frac{q}{p})^z - 1}{(\frac{q}{p})^M - 1}\)
\(q_z = \frac{(\frac{0.6}{0.4})^1 - 1}{(\frac{0.6}{0.4})^8 - 1} =\) 0.0203013.
There is a ~2% probability Smith will win using this strategy.
Using the formula \(q_k = p\cdot q_{k+1} + q\cdot q{k-1}\):
\(q_0 = 0\)
\(q_1 = (0.4)q_2 + (0.6)q_0\)
\(q_2 = (0.4)q_4 + (0.6)q_0\)
\(q_4 = (0.4)q_8 + (0.6)q_0\)
\(q_8 = 1\)
\((0.4)^3 =\) 0.064.
This is a 6% chance of getting out with this strategy
The strategy that allows him to bet the least amount of times nets him the best chance of getting out of jail and that is the second strategy, to bet as much as is necessary to reach the bail amount