Segment 3 - Nick Wilson

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To Calculate

Problem 1

The slides used a symmetry argument ("relabeling") to simplify the calculation. Redo the calculation without any such relabeling. Assume that the doors have big numbers "1", "2", and "3" nailed onto them, and consider all possibilities. Do you still have to make an assumption about Monty's preferences (where the slide assumed 1/2)?

I consider 6 scenarios based on which door I pick and which door Monty opens. Note that I do not consider the scenario where Monty opens the door I chose since the problem states he will never do this. The full calculations are shown for scenario 1 but are abbreviated in the other scenarios because they are very similar.

Scenario 1: Pick door 1 (D1), Monty opens door 2 (O2):

Scenario 2: Pick door 1 (D1), Monty opens door 3 (O3):

Scenario 3: Pick door 2 (D2), Monty opens door 1 (O1):

Scenario 4: Pick door 2 (D2), Monty opens door 3 (O3):

Scenario 5: Pick door 3 (D3), Monty opens door 1 (O1):

Scenario 6: Pick door 3 (D3), Monty opens door 2 (O2):

Under the assumption of Monty's 1/2 preference, the results are the same as with the relabeling technique: I double my chances of getting the car if I switch doors. The 1/2 assumption was still required to get the same numbers as the lectures.

Interestingly, the "always switch doors" strategy is never a bad idea. Consider scenario 1, for example, where I originally pick the door with the car and Monty opens door 2 with probability . If this probability was 1 instead, it would bring the probability of H1 up to the same probability as H3. While switching doors would provide no benefit, it would not hurt my chances.

To Think About

Problem 1

Lawyers are supposed to be able to argue either side of a case. What is the best argument that you can make that switching doors can't possibly make any difference? In other words, how cleverly can you hide some wrong assumption?

I would explain to the contestant that Monty knows the "secret" is out that contestants should always switch doors. Because of this, Monty has adjusted his strategy and now it makes more sense to stay with your original door.

Problem 2

We stated the problem as requiring the host to offer the contestant a chance to switch. But what if the host can offer that chance, or not, as he sees fit? Then, when offered the chance, should you still switch?

Problem 3

Mr. and Mrs. Smith tell you that they have two children, one of whom is a girl.

(a) What is the probability that the other child is a girl?

Mr. Smith then shows you a photo of his children on his iPhone. One is clearly a girl, but the other one's face is hidden behind the family dog, and you can't tell their gender.

(b) What is the probability that the hidden child is a girl?

If your answers to (a) and (b) are different, explain why there is a difference.