MacGougan at Large

Notes on Paradoxes - 2 - Monty Hall Paradox

Monty Hall Paradox

What’s known as the Monty Hall Paradox wasn’t actually devised by Monty Hall. It was devised by a guy named Steve Selvin. The paradox has to do with the three-door challenge that ended every episode of the show “Let’s Make a Deal” that for many years was hosted by Monty Hall.

A desirable prize is hidden behind one of three doors. The contestant is asked to guess - Door One, Door Two, or Door Three. After they choose, one of the remaining doors is shown to be not the correct answer. The contestant is then given a choice: stay with their initial choice or switch to the other remaining option. 

 There are powerful psychological currents that make it difficult for contestants to change their guess at this point. They don’t want to appear fickle. They can imagine how painful it will be to have had the correct answer in their grasp and then traded it away.

Meanwhile, there doesn’t seem to be any compelling reason to switch one’s guess. There are two options remaining. It appears to be a fifty-fifty proposition.

But it’s not. The contestant will double their chances of winning if they change their guess. Technically, this is not so much a paradox as simply a situation that’s counterintuitive. It was certainly not self-evident to me. I had to run several simulations before I became convinced.

The contestant’s original choice has a one-in-three chance of being correct. Because Monty Hall always eliminates an incorrect choice, the remaining door now has a two-in-three chance of being correct.

Some people assume that the Monty Hall Paradox teaches us that we should always say Yes to any last-minute opportunity to change our mind. Reverse the bet, book a different flight, accept a different job offer. But a key element of the premise is that the eliminated door isn’t chosen at random. Monty Hall knows where the prize is hidden and always eliminates an incorrect door. There are very few situations in real life that fit that premise. 

Also, imagine how painful it will be to have had the correct answer in your grasp and then traded it away.