Got a stats question for you, /sci/. It'd be easy if the probabilities didn't change.
My question is, with the following problem, is there a statistical method to solve it, besides exhaustion?

You're playing a game, and you are trying to see if you either succeed at 3 random rolls first, or fail at 3 random rolls first. If at any time you pass 3 rolls, you win. If at any time you fail 3 rolls, you lose.
However, each time you fail a roll, your chances of success go down. Passing a roll keeps your chances of success where they are.

For the first roll, your chances of winning are as follows
>Win: 0.31744
>Lose: 0.68256

Should you fail one roll, your chances of winning drop to
>Win: 0.1792
>Lose: 0.8208

Should you fail two rolls, your chances drop to
>Win: 0.064
>Lose: 0.936

For example, your chances of losing after getting a success, a fail, a success, a fail, then another fail, is about
(0.68256) * (0.31644) * (0.1792) * (0.8208) * (0.936) =~ 0.0297361

If the probabilities didn't change this would be a simple Poisson or maybe negative binomial distribution. The changing probabilities is throwing me through a loop. Can you help, /sci/?