Suppose that you are a contestant on Let’s Make a Deal. You are shown three doors and are told that behind one of them is a car. The other two conceal goats. You choose one of the three doors. Without revealing what is behind the door you chose, Monty Hall has Carol Merrill (I know she left the show a long time ago, but work with me here — she was on the show when I was a kid) open one of the other two doors, revealing one of the goats. Our host now gives you the option of either sticking with your door or of switching to the other unopened door. Which option should you choose to maximize your chances of winning the car?
This issue is called the “Monty Hall problem” (from the title of an analysis of the problem in the journal American Statistician in 1976), and it fools just about everyone. The intuitively obvious answer — that it makes no difference whether you stick or switch — is incorrect. There is actually a big advantage to be gained from switching doors when permitted to do so. The answer is thus a “veridical paradox” because the result appears odd but is demonstrably true nonetheless. You can test it out by playing the game yourself here.
Why should you always switch to the other door? If the car is initially equally likely to be behind each door, a player who picks Door #1 and doesn’t switch has a 1 in 3 chance of winning the car while a player who picks Door #1 but then switches has a 2 in 3 chance. Monty removed an incorrect option from the unchosen doors, so contestants who switch double their chances of winning the car. Put another way, when you stick with your original choice, you’ll win only if your original choice was correct, which happens only 1 in 3 times. If you switch, you’ll win whenever your original choice was wrong, which happens 2 out of 3 times.
Even with this explanation, many refuse to believe that switching is beneficial. After the Monty Hall problem was described and the answer explained in Marilyn vos Savant’s “Ask Marilyn” column in Parade magazine in 1990, approximately 10,000 readers wrote to the magazine claiming that vos Savant was wrong. It’s a common response. Even when given explanations, simulations, and formal mathematical proofs, many people (including many mathematicians and scientists) still do not accept that switching is the best strategy (you can read several funny and arrogant responses from academics to vos Savant here).
The problem can get murkier because Mr. Hall is in control of the game and can play with contestants’ minds. Implicit in the problem’s formulation is the idea that Monty knows where the car is and will always have Carol open a door with a goat behind it after your choice. But he doesn’t really have to. He can even exert psychological pressure to influence your decision.
Suppose your original choice (Door #1) is correct but, understanding the probabilities and not knowing you’re right, you decide to switch doors (to #2) when asked. But further suppose Mr. Hall then offers you money not to switch — perhaps a lot of money. Odds are you’ll still switch even though a bird in the hand should be worth two in the bush. As Monty explains:
“Now do you see what happened there? The higher I got [the more money offered not to switch], the more you thought the car was behind Door 2. I wanted to con you into switching there, because I knew the car was behind 1. That’s the kind of thing I can do when I’m in control of the game. You may think you have probability going for you when you follow the answer in her column, but there’s the psychological factor to consider.
“…If the host is required to open a door all the time and offer you a switch, then you should take the switch,” he said. “But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood.
“My only advice is, if you can get me to offer you $5,000 not to open the door, take the money and go home.”
As with the markets, our psychological make-up tends to push us to “buy high” and “sell low” — to act against our own interests.
As I noted yesterday, we simply aren’t very good at analyzing probabilities. As Persi Diaconis, a former professional magician who is now a Harvard University professor specializing in probability and statistics puts it, “Our brains are just not wired to do probability problems very well….” This issue is of particular import to investing since it always involves analyzing probabilities in uncertainty.
Bottom line: we need to rely on the data, even when it’s both counterintuitive and difficult.