Speaking of not knowing what one is talking about, a media blogger I occasionally read (and have been frequently infotained by for many years) was for an extended period a while back writing almost exclusively about how bad popular math and science books are … because he doesn’t understand them. I was shocked one day to read in the comments of one of his diatribes on the quality of a particular biography of a 20th century mathematician that someone remembered that the blogger had had trouble with the Monty Hall problem years ago (2 administrations ago as it turns out).
For anyone who does not know, the Monty Hall Problem (named for the host of a wacky game show of my youth called Let's Make a Deal) goes in a nutshell like this: You are a contestant on Let's Make a Deal. Monty Hall shows you three doors on stage, behind two of which are goats-- but let's say piles of goat manure to remove an incentive for animal lovers-- and behind the other, in the original framing is a brand new car, but to make it extra interesting, let's say $100,000,000.
You are instructed to select one of the doors. Whatever is behind it is your prize. All things being equal, you have one chance in three of picking the door hiding the money. At this point, one door is just as good as any so you pick one--let's say Door Number 1-- and announce your selection. Instead of simply opening the door you picked, Monty opens one of the two remaining doors (let's say Door 3) to reveal a pile of goat manure. He then offers (in a suspiciously assertive way as was his wont) to let you switch your pick from Door 1 to Door 2. Should you switch?
Think about that for a moment. Remember there's $100,000,000 at stake. I'll wait. < Unloads dishwasher ... Paints house ... Reads Piketty's Capital in the 21st Century >
Ready? The spoiler follows. The blogger, in pretty typical fashion for initiates to the problem, insisted that on having your choices reduced to 2 by Monty Hall (i.e., from Doors 1, 2 and 3 to merely Doors 1 & 2), you actually now had a 50% chance of being right so you might as well resist the pressure and stay put with Door 1. This is the choice that contestants on the actual game show made all the time. As it turns out, and as Monty Hall well knew, it is not correct. The rather surprising answer is that you should switch because 2 times out of 3, when you switch you get the money. Our blogger, in light of the fact that the problem was being posed to him and the solution explained in the pages of the New York Times, not only strongly disagreed with the solution but confidently skewered the paper for spreading disinformation yet again.
What surprised me about the comment that I came across in the recent post was that on the day in question all those years ago, I actually helpfully emailed the blogger (very uncharacteristically on my part) to try to explain why switching when Monty Hall opens a door with a goat behind it paid off with a car 2/3 of the time. The very next day, the blogger acknowledged (very uncharacteristically) that he’d received emails that suggested he might need to re-evaluate the situation, but he doubled down on his assessment based on the New York Times statement of the problem which he was originally critiquing. (He was still wrong!) I don't recall that it was ever mentioned again, and I thought I was the only person to remember the incident, but apparently there are traces of it still out there on the web—not just on the blogger's site.
My explanation went something like this. Imagine instead of 3 doors, there are 4-- goat manure behind 3 of them and the fortune behind one. In this case, you have a 1 in 4 chance of picking the right door the first time. After hearing your choice, instead of one door, Monty reveals goat piles behind 2 of the remaining doors. Now how confident are you in your first choice? Do you still think your first choice jumped from 25% to 50% likely to be correct?
Not yet convinced? Let's say there are 1000 doors. 999 of them have goat manure piled up behind them and 1 has $100,000,000 behind it. You have 5 seconds to pick and nothing to lose so you pick door number 451. Now Monty on hearing your selection, opens 998 of the remaining doors to reveal goat manure leaving only your choice and door number 778 closed. There are once again only two choices left, but are you still confident that the original odds that door number 451 has the money behind it went from 0.1% to 50.0%?
The factor that your mind edits out with 3 doors is that Monty Hall knows where the money is. His choice of door to reveal manure behind is never random. He'll never randomly show you the money before offering to let you switch. The reveal is done to fool you into thinking your odds of having chosen the right door on the first try have increased. In truth they've never changed. What changes is a stark reframing of the act of choosing on the second try. By having manure revealed behind one of the 2 unchosen doors, your odds do not change, but your choice is simplified to Switch or Don't. If you feel confident that your first choice was correct, stick with your first choice. If you feel it's more likely that your first choice was incorrect, change your choice. Staying with your first choice, you don't always lose, but by straightforward probability you do lose 2 out of 3 times whether Monty takes one of the doors out of play or not. Switching to what Monty has conveniently made the only other option on your second chance, you win 2 out of 3 times-- twice as often as staying put.
What have we learned? For one thing, we've learned that when Monty Hall gives you a chance to change doors, no matter how many doors you have to choose from and even no matter how many other doors hiding manure he reveals before asking for your decision, you're always at least somewhat better off switching. Moreover, if I have done my job, I hope we've all learned and can take to heart that contrary to all other evidence day in and day out from time immemorial, on one day in 2006, the New York Times was actually correct about something.
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