You are a contestant on a game show. The host shows you three closed doors and tells you there is a valuable prize behind one of the doors, while there is nothing behind the other two doors. You are asked to pick one door. You will win the prize if it is behind the door you have selected. You pick one of the doors and tell the host which one you have selected.
The host walks up to one of the doors that you did not pick. There is a drumroll. The suspense builds. He opens the door. There is nothing behind it! You breath a huge sigh of relief. It is now down to only two doors, and the prize is behind one of them. Then the host turns to you and gives you a choice. You may keep the door that you picked first, or you may change your mind and pick the other unopened door. Decisions, decisions, decisions. You ponder this for a while. It doesn't really seem to matter which of the doors you pick. The prize is behind one of them. It seems like chances are fifty-fifty either way. To change doors, or not to change doors, this is the question.
The question is, should you change doors, or should you stick with the first door you picked?
More precisely:
1) What is the probability that you will win the prize if you stick with the
first door you picked?
2) What is the probability that you will win the prize if you change your
mind and pick the other unopened door?