21 and the Monty Hall Paradox

Bringing Down the House by Ben Mezrich is, so far as I know, the only book which has ever succeeded in writing about the game of blackjack in an interesting way.

No offense to blackjack authors/players, but blackjack suffers from the same problem that afflicts poker: it can be a lot of fun to play, but often not much fun to read about playing. As an old blackjack-pro-turned-poker-player once wrote:

Blackjack is a game of pure numbers and rote, algorithmic strategy, and the life of a professional blackjack player (as professional blackjack players will agree) can be an exceedingly dull grind. Why should anybody contend with huge variance relative to a measly 1-2% gain, hostile casino staff, and hours of never-ending boredom? Masochism? For this reason, I believe that inside every casino blackjack player is a poker player, waiting to get out. In poker, the edge is a fat 10%, 15%, 20% by some estimates. In poker, there is no hostile casino staff, only people who are glad you showed up to play. In poker, you can make more money in a year than many people will make in a decade, and you can do this even if you’re not a world-class player. Compared to blackjack, the game of poker is like a breath of fresh air.

For this reason, I think Bringing Down the House is a work of real genius. It does the impossible: injects life back into the game of blackjack. Even if the story is a fraud, as a recent article in the Boston Globe questions: who cares. The story is what sells. And this story sold so well that it was turned into a major motion picture: 21.

21 tried to do for blackjack what Rounders did for poker. And okay, it failed. But from this forgettable jumble of poorly written, poorly acted scenes, we can extract one gem. It’s the scene in which Kevin Spacey presents our hero with a puzzle:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

Welcome to the Monty Hall problem, which is admirably explained in layman’s terms here, and demonstrated using Bayesian maths here.

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The correct answer is: yes, by switching doors your chance of picking the “right door” goes from 33% to 66%, provided that the host knows which door contains the car. If the host doesn’t know which door contains what and is just picking randomly, then the answer is no: your chances of picking the right door are 50-50.

And if that makes zero sense to you…welcome to the club. As Wikipedia notes:

When the problem and the solution appeared in Parade, approximately 10,000 readers, including nearly 1,000 with Ph.D.s, wrote to the magazine claiming the published solution was wrong.

This problem is still vigorously debated today. In Monty Hall Redux, Brian Hayes explains:

In the July-August issue of American Scientist I reviewed Paul J. Nahin’s Digital Dice: Computational Solutions to Practical Probability Problems, which advocates computer simulation as an additional way of establishing truth in at least one domain, that of probability calculations. To introduce the theme, I revisited the famous Monty Hall affair of 1990, in which a number of mathematicians and other smart people took opposite sides in a dispute over probabilities in the television game show Let’s Make a Deal. (The game-show situation is explained at the end of this essay.) When I chose this example, I thought the controversy had faded away years ago, and that I could focus on methodology rather than outcome. Adopting Nahin’s approach, I wrote a simple computer simulation and got the results I expected, supporting the view that switching doors in the game yields a two-thirds chance of winning. But the controversy is not over. To my surprise, several readers took issue with my conclusion. (You can read many of their comments in their entirety here.) For example, Bruce Sampsell of Chapel Hill, N.C., wrote:

If you don’t know who to believe, try an interactive version of the puzzle such as the one below by Shawn Olson:

I was just recently introduced to the Monty Hall Game paradox by my friend Andrew Penry. When he first proposed the game to me I thought it was simply absurd—and my intuitive thinking process would not allow me to accept the statistical conclusions that the game entails.

Sit down and play 50 or 100 games, and see whether you end up choosing the right door (or in this case, the right card) 66% of the time, give or take. Here’s how my results looked after 60 games:

That’s a highly unintuitive result from the perspective of anyone who’s been trained to think that past results don’t change the probability of independent future results. Flipping a coin and getting heads 10 times in a row doesn’t change the probability of getting heads on the next flip. Your chances are 50-50, every single time. Similarly, just because the host has revealed 1 of the 3 doors doesn’t change the fact that we’re now presented with a choice between two doors. One door contains a car and one door contains a goat. Ergo, no matter which door you choose, and no matter what happened previously, your chances of picking the right door are now 50%.

That’s rational, makes perfect sense, and yet, like many things in life which are rational and make perfect sense, it’s 100% wrong.