@virtualist; the probability of winning on the first pull isn’t even 60%, it is 10% + 15% = 25%. No, LostInParadise asked what he meant to (he tends to do that). The question can be restated as “given the fact that there are two machines, one of which (A) wins 20% of the time and the other (B) 30% of the time, and the fact that when one of the machines is played once it returns a win, what is the chance that this machine is machine B.”
If I were to ask, what is the probability that I was born on April 12, you would say 1/ 365 (unless you’re a jerk who remembers leap years). But if I also gave you the fact that I was born on the 12th of some month, you would say the chance is 1/12. And then, if I told you I was born on April 12, you would say the chance is 100%. Obviously, the chance was always 100%, but your previous answers were still correct, because when we talk about probability, we are just talking about how certain we can be that something is true, given what we know. Therefore, the fact that we gain the information that the machine returned a win, will obviously improve the probability that the machine we chose is the one with a greater tendency to win.
Technically, this is the exact same logic that allows us to say that what looks like a horse, acts like a horse, and smells like a horse, is probably a horse; the chance that a random organism is a horse is very small (and is analogous to the 50/50 chance of a random machine being machine B), but given a horse has a much higher chance of appearing to be a horse than a non-horse does (analogous to the higher chance of machine B winning than machine A), we can safely conclude that what appears to be a horse (a machine that wins) is probably a horse.
There, I think that should be enough examples for now.