It really boils down to one thing: Your odds after your first pick don't change. On your first pick, you had a 1/3 chance of picking the correct door. After Monty opens up a goat door, your odds that the door you picked has the prize behind it are
still 1/3. There really is no probability of 1/2 that ever enters the equation.
If there were four doors, you picked one, then Monty opened up two goat doors, the odds that the first door you picked has the prize remains 1/4. If there were seven doors originally, the odds are 1/7. For any
n doors, if you pick one, then Monty opens
n-2 doors, leaving just the door you originally picked and one other door, the odds that the door you originally picked has the prize behind it are 1/
n. By the laws of probability, that necessarily means that the odds that the other door has the prize behind it are (
n-1)/
n.
The thing that Guy is overlooking is that he fell into the trap that the problem is designed to make people fall into: to disregard circumstances leading up to a decision, then assuming that all probabilities must be equal. It's really diabolical. There's an old trick question that exploits the converse. It goes something like this: If you flip a fair coin 587 times and it comes up heads 587 times straight in a row, what are the odds that it will come up heads on the next toss? The answer, of course, is 50%.
Most people have that pounded into their heads at some point during their mathematical education, and so when a problem such as the Monty Hall problem is presented to them, they automatically mentally make it equivalent. There are two doors, one prize, and thus the odds are 50/50 for each door; it doesn't matter which you pick. Guy actually says this explicitly in his OP:
[10:00:16 PM] Guy Perfect: You said there was one prize.
[10:00:18 PM] TonyV: True.
[10:00:22 PM] Guy Perfect: And two doors.
[10:00:25 PM] TonyV: Yes.
[10:00:32 PM] Guy Perfect: Which means one of two doors has a prize.
[10:00:37 PM] TonyV: Yes.
[10:00:42 PM] Guy Perfect: Ta dah! 50%.
[10:00:46 PM] TonyV: Nope!
*spring! He fell into the trap. There was no miscommunication, there was no misunderstanding, there was no, "I thought you meant
x and so I answered that way." I didn't misrepresent the problem, I didn't screw up my presentation of it in any way. There weren't two scenarios.
He really shouldn't feel bad.
According to the New York Times, "The experts responded in force to Ms. vos Savant's column [in which she presented the Monty Hall problem]. Of the critical letters she received, close to 1,000 carried signatures with Ph.D.'s, and many were on letterheads of mathematics and science departments."
But of course, we're expected to believe that Guy is a mathematical genius who didn't fall into the same trap that people with mere Ph.D.s fall into. No, he came to the
exact same conclusion--even for the
exact same stated reason ("Well, my brain is telling me that since I still know nothing about doors 1 or 3, there may as well be an even 50/50 for it being the prize") because he simply misunderstood the problem, which was perfectly stated.
But the thing that he and other people aren't taking into account is another little fact: Just because a coin comes up heads 50% of the time after several tosses does
not mean that the coin is, in fact, a
fair coin! If one side, for example, has been weighted to be significantly heavier, it could shift the probability of it coming up heads to, I dunno, let's say, 1/3. Two possible outcomes,
not 50/50. If you bet on heads, you're probably going to lose.
That's exactly what's happening with the Monty Hall problem. A sequence of events leads up to a choice between two options that are
not 50/50, but it cleverly disguises that fact by presenting a specific sequence that, on the surface, doesn't seem to matter much.
What I find amusing is that at this point, Guy either 1) made a mathematical error that experienced mathematicians, including many Ph.D.s have made, or 2) made a reading comprehension error that a fifth-grader would have nailed. In his extraordinary effort to try to sound smart, he is using the excuse of doing something ten times dumber. He
thinks that as long as he has technically not admitted that he was wrong, everyone will somehow magically buy it.
Personally, I don't mind admitting I was wrong. I've done it before, and I'll do it again. I mean, I don't relish in it or anything, and I do diligently try to be right most of the time, and in fact, I'm a pretty bright guy who is right most of the time. But I won't hesitate to say that when I first encountered the Monty Hall problem, I screwed it up in the exact same way that Guy did. I was talking about it with a friend, and like Guy, I tried six ways to Sunday to convince my friend that he was utterly and completely wrong. But once I got it, I got it. I admitted I was wrong, he had a good laugh at my expense, and I added it to my repertoire of really cool and clever brain teasers.
Meanwhile, it's two days later, and Guy still insists that he was right. Dude, if your goal is to look smarter, it ain't working too good. Cut your losses already; you'd probably be surprised how much more respect admitting your wrong garners when you've been gotten than making such a fuss like this.
