Mary: Damn it, I thought I might catch some people with the apples puzzle, because after the cups puzzle people wouldn't expect a simple solution...

Murray: I heard the cups problem from my macroeconomics professor. I can't remember what connection he made between it and macroeconomics. The red/white thing does seem related at first blush, but I'd like to think about it some more before I comment. I'm intrigued that Jim's answer was so helpful to you; while it's nice to have someone agree with me, I couldn't understand how his explanation made it clearer. The thing that fascinates me about this problem is that the wrong answer is so simple and obvious whereas the right answer takes a lot of persuasion and there are a number of lengthy, counter-intuitive ways to explain. There's probably some lesson about life here.

Andres: suppose you played this game several times, and suppose that you believed it didn't matter whether you kept your first guess or switched... in that case you would be picking randomly between the two cups, and you would win approximately 50% of the games. In this scenario, your second guess really would be an independent event with no connection to your first guess, so your comparison of this to one toss in a sequence of coin tosses is reasonable.

However...as I said, by picking randomly between the two, you will win about 50% of the time. But if you look back at the games you win, you will find that in most cases your second guess was different from your first. And if you examine the games you lose, you will find that in most cases your second guess was the same as the first. Therefore, you can improve your chances of winning by changing your guess.

If you don't believe me, you can confirm it with the formula for calculating the probability of a complex event that you mentioned in your post. If you write down every possible outcome in this game, then for each possible outcome multiply the probabilities of each independent event in the complex event.

For example: P(prize is in cup 1)=1/3
P(your first guess is cup 2)=1/3
P(house lifts cup 3)=1
P(your second guess is cup 1)=1/2
Probability of this outcome as a whole: 1/18

Then, if you add up the probabilities of the outcomes where the player changes guesses and wins the game, the total will be twice the total probability of the outcomes where the player guesses the same cup twice and wins the game. I've done this table and done the math (yes, I really am that obsessed) and I would say that this is irrefutable evidence according to the laws of finite math. Unfortunately I'm not an HTML expert so an attempt to produce a table here would be chaotic I'm sure, but do the table and the math yourself if you want.

Thanks for playing, Marion