Think and Check!

Student: The problems like Monty Hall and the three box puzzle are designed to get on people's nerves! You would think the answers are obvious, but the experiments give different numbers!

Mentor: In the Three Boxes game, each box contains two chips: the first has two red chips, the second has two green chips, and the third has one red and one green chip. We do not know which box contains which chips. We take one chip out of a box without looking inside, and it turns out green. What are the chances, theoretically speaking, that the second one is also green?

Student 1: There is only one box out of three that has two green chips, so the probability should be 1/3.

Student 2: We know that the first chip was green. There are only two boxes for which it is possible. One of them has two green chips, so the probability is 1/2.

Student 3: Wait, we have played the game on the computer many times. The experimental probability was getting closer and closer to 2/3.

Student 4: It makes sense: there are three situations when the first chip is green: it may be either one of the two chips from the box with two green chips, or the green chip from the box with one green and one red chip. In two of these situations, the second chip in the box is also green, so we have two winning outcomes out of three.

Mentor: What about the Monty Hall problem? The game simulates a well-known game show situation that used to happen on the Monty Hall game show. A player is given the choice of three doors. Behind one door is the Grand Prize (a car and a cruise); behind the other two doors, booby prizes (pigs). The player picks a door, and the host peeks behind the doors and opens one of the remaining doors. There is a booby prize behind the open door. The host offers the player either to stay with the door that was chosen at the beginning, or to switch to the remaining closed door. Which strategy (to switch or to stay) gives you a better chance of winning in the long run?

Student 1: All the doors are the same, so when only two doors with one prize are left, the chances of winning when you select any one of them are the same: 1/2.

Student 2: The probability of a prize being behind any one of the three doors is 1/3, so it does not matter whether you switch or stay: the chances of winning will be 1/3.

Student 3: But the experiments give us the probability of winning of 1/3 with staying and 2/3 with switching. It makes some sense: when you choose a door at the beginning, the probability of picking the prize is clearly 1/3, and the probability of the prize being behind the other two doors is 2/3. When Monty opens one of the other doors, the probability of 2/3 "concentrates" in the second door that remains closed.

Mentor: It is always a good idea to check your theory with experiments, when possible. Luckily, it is quite easy to do for most probability problems if we can use computers.