Let's try looking at the problem backwards to see if we learn anything.

Say we have 2 cups. One of the cups is on your right and one on your left.

Prior to your walking into the room, I have placed a gold coin beneath one of the cups. I tell you that l rolled a "fair" die and if it came up 1 or 2, I put the coin under the left cup; otherwise, I put it under the right one.

So if
L = coin is under left cup, and
R = coin is under right cup, then we conclude:

P(L) = 1/3 and P(R) = 2/3.


Question 1: Does this conclusion make sense? If not, then don't proceed to the next part.

Note: If you have gotten this far, it means you realize very clearly that the probability of the coin's being under each of the two cups is not the same.

Unfortunately, while I was distracting you with that question, someone decided to be clever. He moved the cups around and around and around. You turn your head back suddenly and you see the cups are jostled from their original positions.

You can no longer tell which cup is which.

What now is the probability that the coin is under the left cup?

I assert that P(L) = 1/2 and P(R) = 1/2.

Question 2: What is different about this situation from the initial state?

Answer: We have lost information.

Conclusion: Our calculations of probability are based on how much information we have about the situation.

Gaining or losing information changes the probability.