Weird Probability (but a bit simpler)

Hello all,

I posted earlier in the day (please forgive this), but I made the problem a bit simpler so I thought I would give it another go. Basically, I cannot figure this out for the life of me, and I am starting to think it may not be solvable (but I hope you people who are much smarter than me can shed some light).

So I want to know the expected value of a prize y (I am comparing this with a lottery in which you get a prize for certain). At time zero, you have a probability (1-x) of getting y, and a probability of x of getting 0. If you got the prize y at time zero, you have a new probability (1-2x) of getting the prize at time 1 and a probability 2x of not getting it.

So the tricky bit is, if you did not get the prize at time zero, the probability of getting the prize at time 1 remains the same (that is, 1-x). So this goes on for an infinite number of periods... The probability of getting the prize goes down if you got the prize yesterday, but it stays the same if you didn't.

I hope this makes sense, and really any help (even to put me out of my misery) will be appreciated more than you probably know.

Sincerely,

Nick