Another Golden Afternoon

Advisory: contains marginally geeky content. Reader discretion is advised.

I’m back at Boalt for my third year of law school. Just before I moved back to the East Bay, though, I had the pleasure of catching up with enji for a day. We don’t get to hang out much, certainly not compared with before, and one-on-one time is becoming increasingly rare and precious. I spend a lot of time studying, and enji spent most of her time this year out of town. Sushi and tea in Mountain View, for about four hours or so.

We spoke of tea, of law, of music, of spliced movie titles, of expectations and family relationships - indeed, we could probably talk about almost anything and have a good time of it. Such is a conversation with Enji - there is always something to touch the soul, to tickle the wit, and sometimes, a sudden and unexpected or abstract observation that gives a moment's pause for consideration. "Wow. Hmm, give me a moment to think about that." We ended up spending about an hour and a half of that time talking about statistical probability. Very geeky, yes, but very fun; admittedly, we both have backgrounds in computer science, which explains the geek factor, but leaning back and looking at the discussion from a third-person perspective inspires a fit of girly giggles. (Without knowing more, the thought of an artist and a law student trying to figure out a puzzle of probability and statistics strikes me as a case of the blind leading the blind. Some of my fellow law students freak out when faced with the task of taking the average of three whole numbers.)

I vaguely remember coming around to the topic of the Monty Hall Paradox (an elegant problem of statistical probability if ever there was one) after playing a few rounds of various card games (always a handy segueway into the subject of probability). Mostly, I’m writing this down so I don’t forget how we came across the answer to this puzzle. It’s a nontrivial puzzle with a counterintuitive answer; I make no claims to being particularly clever, since reverse-engineering the explanation from a known answer is usually a heck of a lot easier than coming up with the observation in the first place. Still, I’m a little jazzed by the fact that we figured it out, so I’m posting the answer to myself so that I won’t forget it. (I sure didn't get it the first time around, about 9 years ago. It's been nettling me somewhere in the far reaches of my subconscious for all that time, I'm certain.)

The Monty Hall problem arises from the old “Let's Make a Deal” game show, where the host, Monty Hall, presents the contestant with three numbered doors and invites him to choose one. One of the doors has a prize behind it; the other two doors conceal (frequently undesirable) gag gifts. The contestant, of course, has a 1/3 chance of picking the door with the good prize. But after the contestant has made his choice, Monty opens one of the other two doors (one concealing one of the gag gifts. If you take any two of the three doors, at least one of them conceals a gag gift.) Monty then asks the contestant if he wants to change his mind about which door he’s chosen. With one of the three doors eliminated, the answer that suggests itself is this: the contestant has a 50% chance of winning, whether he sticks with his original door or picks the other one. However, the real answer is this: if the contestant stays with his original choice, his chance of winning remains 1/3, and if he switches, his chance of winning rises to 2/3. The answer to this problem apparently confounded a number of esteemed and capable mathematicians and academics.

My memory of the problem was less than perfectly solid (hey, I didn’t really get it the first time around, after all), and so we start drafting decision trees and combinatorics into enji’s travel notebook (which already contained a pretty interesting mix of practical reminders, travel notes, and word games).

Basically, the Monty Hall paradox is understood by realizing the actual effect of Monty’s revelation of one of the bogus prizes. The information Monty reveals after the player’s original choice inverts the outcome of the player’s choice. If the contestant had chosen the correct door to begin with and switches, he loses. If he or she picked one of the wrong doors, though, Monty’s choice ensures that the contestant will win if he switches. The 1/3 original win turns into a 1/3 loss, and the 2/3 original loss chance becomes a 2/3 win chance. If the player pre-commits to a strategy of always switching, he or she can take advantage of Monty’s inversion of the win probabilities.

If the player picked the right door to begin with (a 1/3 chance), he will lose. Why? Monty can reveal either one of the bogus doors; it doesn’t matter. The player’s switch will be away from a winning door to a losing door.

But if the player picked one of the wrong doors to begin with (a 2/3 chance), Monty’s actions are constrained; by the rules of the game, he can’t reveal the winning door, only a losing door. Since the contestant is tentatively standing on one of the losing doors, Monty must knock out the other losing door. By switching, the player moves from a losing door to the winning door.

And that’s how it works. The rules of the game cause Monty’s information to perfectly, without exception, reverse the original outcome, turning wins into losses and losses into wins if the contestant has committed himself to switching doors after Monty’s information.

After we worked out the reasoning, I plopped back into the chair, lifted my teacup, and dryly mumbled “another golden afternoon.” But really, there’s still some of the geek left in each of us, so I have to confess that I really did enjoy the mental exercise, and I think enji got a kick out of it too.