Think Bayes, one of the books by Allen B. Downey, in chapter one, he mentioned an example known by the name
The Monty Hall problem :
The Monty Hall problem is based on one of the regular games on the show. If you are on the show, here’s what happens:
Monty shows you three closed doors and tells you that there is a prize behind each door: one prize is a car, the other two are less valuable prizes like peanut butter and fake finger nails. The prizes are arranged at random.
The object of the game is to guess which door has the car. If you guess right, you get to keep the car.
You pick a door, which we will call Door A. We’ll call the other doors B and C.
Then Monty offers you the option to stick with your original choice or switch to the one remaining unopened door.
Before opening the door you chose, Monty increases the suspense by opening either Door B or C, whichever does not have the car. (If the car is actually behind Door A, Monty can safely open B or C, so he chooses one at random.
He then explains the problem, that we are supposed to understand :
To start, we should make a careful statement of the data. In this case, D consists of two parts: Monty chooses Door B and there is no car there. Next we define three hypotheses: A, B, and C represent the hypothesis that the car is behind Door A, Door B, or Door C. Again, let’s apply the table method :
Figuring out the likelihoods takes some thought, but with reasonable care we can be confident that we have it right:
If the car is actually behind A, Monty could safely open Doors B or C. So the probability that he chooses B is 1/2. And since the car is actually behind A, the probability that the car is not behind B is 1.
If the car is actually behind B, Monty has to open door C, so the probability that he opens door B is 0.
Finally, if the car is behind Door C, Monty opens B with probability 1 and finds no car there with probability 1
What I don't understand is How is
p(D|H) = 1 in
C? Because Monty could also open door A. So the
p(D|H) should be equal to
C. Also, I don't think I fully understand the reasoning behind case A and B either. Can someone explain this in more details ?