Toot has T pennies. Sly has S pennies. Toot wins when two coins are tossed and they both come up heads or both tails. Sly wins when the coins come up with one being a head and the other a tail. These are all fair toss coins.
p(n) denotes the probability that Toots wins all of Sly's coins if Toots starts with n pennies.
First write a recurrence relation for p(n)
Second Solve this relation
Third Find the probability that Toot wins all of Sly's pennies
I have the answer to the first
p(n) = 2 p(n-1) - p(n-2)
My problem is I do not understand how they came up with
p(n-1) = 1/2p(n) + 1/2p(n-2) once I am at this point it is easy to get the p(n) relation but how do you think of this equation for p(n-1)???
Then I need help to solve it. I know there are n coins to start with
and I was thinking I should use
p(n) - 2p(n-1) + p(n-2) = 0
Then t^2 - 2t + 1 = 0
t=1 and only one
I am not sure how to proceed and once I have equation I do not know how to come up with a number for the probability???