Question 1

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 at a time.

The equation for this type of problem is as follows: q = lose = 0.6 p = win = 0.4 M = steps to end state = 8 z = start state = 1

P(lose) = ((q/p)^M − (q/p)^z)/((q/p)M − 1) = (0.6/0.48-0.6/0.41)/(0.6/0.4)^8-1) = 0.9797 P(win) = 1-P(lose) = 1-0.9797 = 0.0203

b) he bets more aggressively.

This means if he starts at 1 he would double to get to 8 as fast as possible. 1->2->4->8 this means it would take 3 steps to reach the end state.

M changes to 3.

P(lose) = ((q/p)^M − (q/p)^z)/((q/p)M − 1) = (0.6/0.43-0.6/0.41)/(0.6/0.4)^3-1) = (3.375-1.5)/(3.375-1) = 0.7895 P(win) = 1-P(lose) = 1-0.7895 = 0.2105

c) which strategy should he follow?

he should take the aggressive approach.