Mary, there's always a 50% chance that the next coin toss will be tails. But I don't think that that puzzle has much bearing on my cups game. Each coin toss in the series is an independent event. But in the cups game, your two chances to guess are related to each other; the chance that your second decision will be successful is directly dependent on the the success of your first guess.

The cups game could be considered a series of independent random events if you forgot which cup you had chosen first and simply guessed randomly between the two cups that hadn't been lifted by the house. In that case, you would win about 50% of the time; it would be a question of choosing between two cups, not of choosing between sticking with a first guess that was probably wrong or abandoning a first guess that was probably wrong. But in 2/3 of the games you win, your winning cup would happen to be one that you didn't pick on your trial run guess.

Here's another little puzzle, for free:

Suppose you have a basket that can hold ten apples. You take out three. How many apples do you have?

Marion