A warm up problem 1 Somebody flips two coins on table, and then hides them with some two pieces of clothing. You observe, when one of the coins is revealed, and it is heads or tails. You are then asked to guess what's the side of the unrevealed coin. Which ever answer you give, you are right and wrong with probability 1/2. This is because the unrevealed coin is independent from the revealed one. Right? A warm up problem 2 Two coins are again flipped and they are both hidden with some curtain. You are told that "one of the coins is heads." Now you should guess that the other coin is tails. This way you are right with probability 2/3, and wrong with probability 1/3. This is because in the beginning there was probability 1/4 that both are heads, 1/4 that both are tails, and 1/2 that one is head and one is tails. When you receive the information that one the coins is heads, then the corresponding conditional probabilities are going to be 1/3, 0 and 2/3. Right? The game One hundred pairs of coins are flipped, and each pair is hidden with a curtain, so that we get one hundred "curtain spots". You will go through all of these curtain spots, and you are always told truthfully that "one of the coins is heads" or "one of the coins is tails", and you are supposed to guess what the untold coin is. If you guess correctly, you are given a dollar (or euro), and if you guess incorrectly, you lose a dollar (or euro). A winning strategy We know that if, at every curtains spot, you always guess that the untold coin is different from the told one... meaning that if you are told that "one of the coins is head", you answer "the other one is tails", and if you are told that "one of the coins is tails", you answer "the other one is head" ... then you will always be right with probability 2/3. This means that there is going to be approximately 67 spots where you win, and 33 spots where you lose. You should make approximately 34$ (or 34€) profit with this strategy. But the winning strategy doesn't work! There is going to be approximately 25 spots where both coins are heads, 25 spots where both coins are tails, and 50 spots where one coin is head and one is tails. So with the winning strategy, there is going to be approximately 50 spots where you win, and 50 spots where you lose, the profit will be approximately 0$ (or 0€). How is it possible, that you will only win at approximately 50 spots, out of 100, despite the fact that at every single spot, your winning probability is 2/3?