Coin Probability

A box contains two coins. One coin is heads on both sides and the other is heads on one side and tails on the other. One coin is selected from the box at random and the face of one side is observed. If the face is heads, what is the probability that the other side is heads?

Answer:

Yes, the answer is 2/3. Numerous people have tried to explain why they think the answer is 1/2, arguing that since both coins have a head then seeing a head doesn't rule out anything and thus it could be either coin with equal probability. While it is true that seeing a head does not rule out anything it still provides valuable information that can not be ignorred.
To illustrate why the 1/2 answer is wrong let us propose a similar problem. Suppose a soup company canned 2,000,000 cans of tomato soup. Suppose further that they put the wrong label on half of the cans. The company then sent 999,999 correct cans and 1 incorrect can to one store and 999,999 incorrect cans and 1 correct can to another store. To determine which store has the correct cans you go to one store at random and open one can. You see it does indeed have tomato soup inside. What is the probability you went to the store with 999,999 correct cans? It is not reasonable to assume the probability is 50/50 that you are in either store. The odds that you would pick the one correct can in the store with 999,999 incorrect cans are extremely remote. It is true that before opening a can the odds were 50/50 but once you do open a can that information gives you a clue that can not be ignorred. In fact the probability that you are in the store with the correctly labled cans is 99.9999%.