In the classic movie “It’s a Wonderful Life”, George Bailey and Mary Hatch perfo
ID: 3306313 • Letter: I
Question
In the classic movie “It’s a Wonderful Life”, George Bailey and Mary Hatch perform the Charleston at the school dance, which is held at the gym. Assume that each step of their dance has them take a step to the left with probability 2/3, and a step to the right with probability 1/3. We’ll assume the gym is infinitely wide, and if left to their own devices, the couple would dance forever. Mary and George begin their dance at position x = 3. A prankster, who is jealous that George is dancing with Mary, opens the floor of the gym, which contains a large swimming pool located at x = 0. What is the probability that Mary and George will eventually fall into the swimming pool?
Explanation / Answer
From a position at x, Mary and George can go to x-1 with probability 2/3 and x+1 with probability 1/3. If they reach x=0, they fall into the swimming pool. Hence we need to find the probability that Mary and George go from x=-3 to x=0, i.e. 3 steps to the right eventually.
This problem is similar to a Gambler's ruin where a Gambler starts with i dollars and can win or loss with probability p and 1-p. The probability that he goes eventually broke is 1-(1-(q/p)^i) = (q/p)^i
So, comparing to that problem, we have i=-3, p=2/3 and q=1-p=1/3
The probability that they eventually go from state i to state -inf is : 1-(q/p)^i.
Here we need to go from state -3 to 0 with p=chance of success=2/3 and q=1/3. So, required probability = (q/p)^i = (1/2)^3 = 1/8