Complex Numbers

Hall of Fame

The article I'm sharing today, from SciAm, is a bit esoteric. To me, it illustrates why common sense is neither, and why probability can get really weird.

    See Why Everyone Gets the Monty Hall Puzzle Wrong  
How to finally wrap your mind around the uniquely counterintuitive Monty Hall dilemma

In high school statistics class, my teacher presented a probability problem that haunts me to this day. It was a puzzle inspired by the TV game show Let’s Make a Deal and named after its longtime host, the late Monty Hall.

Several generations of people would never have heard of this guy if it weren't for this puzzle. And it's not like he came up with it; he's just part of the setup. As for the details of the puzzle itself, the article provides a fun illustration of it, but, in short:

There are three doors in front of you.
Behind one of the doors is a new car. Behind the other two doors are goats.
The host invites you to pick a door, any door.
The host throws open one of the doors you did not select, revealing a goat.
Then the host gives you the option of switching your [initial selecton].

Let's leave aside for the moment the relative value of cars and goats. I don't want a new car. I'd be responsible for taxes and insurance on the thing, though I could maybe just sell it for money. But right now, I could really use a goat; you wouldn't believe the condition of my back yard.

Most people think it doesn’t matter whether they stick with their original choice or switch to the other unopened door because the odds are 50–50—that it’s nothing more than a coin toss.

Yeah, that's just common sense. (Which is why I hate "common sense.")

But you should always switch doors. You win two thirds of the time if you switch and one third of the time if you stay. In other words, switching doors doubles your chance of winning.

I have a quibble about "doubles your chance of winning:" If you buy two lottery tickets instead of one (assuming different numbers), that doubles your chance of winning from, I don't know, 1 in a billion to 1 in half a billion. Wow. But, okay, in this particular formulation, it's meaningful.

This counterintuitive problem, first described in relation to the TV show in 1975, is taught in introductory math and statistics classes across the world. But it was widely popularized in 1990 in Parade magazine. After writer Marilyn vos Savant wrote about the puzzle in her Ask Marilyn column, she received an estimated 10,000 furious letters declaring the answer she gave was wrong, including 1,000 or so that were signed by people with a Ph.D. in their title.

It's entirely possible to be smart and wrong. Even if you have a Ph.D. Even if your name is Marilyn vos Savant.

How are the odds not 50–50? Since that fateful column, mathematicians, psychologists and philosophers have been trying to understand what makes this answer uniquely hard to grasp. They’ve found that some of the most common cognitive biases may be to blame, along with a core misunderstanding of how probability works.

The article, of course, goes into the details, and I won't rehash them here. But I will point out another glaring problem we often have with probability, which is almost always illustrated for me when I'm at a blackjack table.

Put as simply as I can, you have the choice in blackjack to stand (keep the cards you have) or hit (dealer gives you another card). The actual probabilities vary, of course, and there can be other complications, but in most cases, you want to either stand or hit, and one of those two has better odds of not losing the game.

But a probability is just that; it's not a certainty. Even if it pays off most of the time (with "most" being anywhere from 50.1% to 99.9% or thereabouts), it won't pay off all the time.

In blackjack, if your strategy fails, you just pony up another bet and play again (or give up in disgust and visit the buffet). But the Monty Hall problem is a one-off. You'll never again be faced with maybe getting a shiny new car or maybe getting a poison-ivy-eating goat, so you only get that one shot. And that one shot is, according to the math, about a 2/3 chance: better than 50-50, but in the end, you might still "lose."

My point being that even the best strategies can fail, and sometimes, we have a problem confusing "probably" with "certainly." And that one data point doesn't mean anything; if you lost on an 80% probability win, that doesn't mean that the next one doesn't have an 80% probability.

Which is why I quit giving probability lessons to other blackjack players.