The Monty Hall problem

A classic brain teaser that shows up in university-level discrete math courses is the Monty Hall problem:

You're on a game show and are presented with three doors. Behind one of the doors is a fancy new car, but behind the others are goats. You're asked to choose a door, and when you do, the host opens one of the doors that you didn't pick and reveals a goat. He then asks, "Do you still want the prize behind the door you've already chosen? Or would you like to switch doors?"

At first glance, it seems like after the host has revealed a goat, the car is equally likely to be behind either of the two remaining doors, so you have a 1/2 chance of picking the correct door. Under this assumption, switching doors is useless.

However, this calculation doesn't take into account the initial odds from the game's layout. In fact, the car is equally likely to be behind any of the three doors, so once you pick a door, there is a 1/3 chance that the car is behind your door and a 2/3 chance that the car is not behind it. Even after the host reveals a goat, those odds don't magically change. There is still a 2/3 chance that the car is not behind your door. Presented this way, you're twice as likely to win the car if you switch doors.

This second calculation has been proven to be correct through various rigorous methods, but it's so counterintuitive that people often refuse to believe its validity. Situations that produce this kind of true but seemingly absurd result are called veridical paradoxes.

The Sleeping Beauty Problem

Here's another fun veridical paradox that I only learned about earlier this week:

Sleeping Beauty agrees to participate in an experiment. On Sunday, she is put to sleep. While she sleeps, a coin is tossed. If it lands on heads, she is awakened and interviewed on Monday only. If it lands on tails, she is awakened and interviewed on both Monday and Tuesday. After each interview, she is put back to sleep and loses all memory of the interview or of being awakened. During each interview, she is asked, "To what extent do you believe that the coin landed on heads?"

This is the Sleeping Beauty problem. Since a fair coin toss is 50/50 and Sleeping Beauty gains no information from the interviews, it's reasonable that she'd believe that the odds that the coin landed on heads is 1/2.

However, once again, this calculation doesn't consider all the available data. Sleeping Beauty is aware of the experiment and knows that she is always interviewed in one of three different contexts:

  1. the coin landed on heads, and it is Monday
  2. the coin landed on tails, and it is Monday
  3. the coin landed on tails, and it is Tuesday

From this perspective, the fair coin landed on heads during only 1/3 of her interviews. So maybe she should believe that the probability that the coin landed on heads is 1/3.

Unlike the Monty Hall problem, there is still ongoing debate about which calculation is correct. All kinds of arguments exist to promote the halfer position vs. the thirder position, but the primary reason that this debate continues is because a statistically rigorous solution hasn't yet been devised and run. So really, the question is: does a rigorous solution exist, and if so, how would it work?

Final (unrelated) thoughts

I stumbled upon the Sleeping Beauty problem while already halfway down an unexpected rabbit hole from flipping a digital coin with a friend. My device's default language is French, and my coin landed on an image that showed a head but was labelled "pile," which is effectively "tails" in English. I eventually learned that the French version of "heads or tails" (called "pile ou face") makes more sense when playing with a euro coin. The "pile" side of a euro coin has the monetary amount written out, while the "face" side has a face or symbol on it. And unfortunately, the digital coin I was using had a head on one side and a symbol on the other.

So all my research didn't end up solving my problem, and I eventually resorted to a physical coin. Still, I thought it was neat to learn about a few new probability problems involving coins – check out the St. Petersburg paradox! – along the way. They were an entertaining surprise that reminded me of my college days.

What about you? What are your favorite paradoxes?