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???