It was a nice illustration for me of feeling my brain rewire itself. It was so counter-intuitive that I was very sure that the odds were 50/50 but then I started thinking about the problem at it’s limit of a lot of doors, and it became intuitive all of a sudden. It was an interesting feeling, one of the only times I’ve been quite sure of something and had to change my mind. However when I ran the same problem by my (old) father, even with a million doors he still couldn’t intuit it, and was totally sure the odds were still 50/50. His brain seemed unable to make the switch. Perhaps an age thing …
the 1,000,000 doors didn’t help me to understand it. What helped me was the illustration in the Wikipedia article showing the expectation of doing this for 300 tries. If you pick door 1, on average, you will have the correct door an average of 100 times. If Monty opens door 2., on average 100 of those times will be because the prize is behind door 3 because that is his only choice. But if the prize is behind door 1, your initial choice Monty can open either door2 or door 3. So the expectation, is that, on average, if your initial choice was correct, he will have opened door 2 50 times and door 3 50 times. And we KNOW that he didn’t open door 3. So that leaves an average of 50 times that your initial choice was right and 100 times that switching to the remaining unopened, unchosen door was correct.
Weird. For me, it becomes very intuitive with a large number of doors that when Monty is opening doors he is giving you new information on where the prize isn’t, and that the door he is studiously avoiding is where the prize probably is. The difference is that with only three doors, the difference in probability is quite small, but with a lot of doors, it becomes obvious that your chance of having picked the correct door is rather low. Then it’s obvious that there is a much higher likelihood that the prize is behind that one door that Monty didn’t open.
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Maybe the odds of feeling better about your choice are higher if you stay. If you choose wrong initially, you never had the prize. If you choose right and switch, you didn’t just not-win, you lost.
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