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Winning a Game Show with Algorithms

Probabilistic reasoning is the heart of machine learning algorithms. For artificial intelligence to break out of the narrow contexts it was used for in the 20th century, it must be able to assign probabilities to events and objectively update those probabilities as new information arises.

While probability theory is in the DNA of machine learning algorithms, probabilistic puzzles often elude humans.

See where your intuitions pull you in this probability puzzle known as the “Monty Hall Problem,” after the infamous game show host and the format of his show, “Let’s Make a Deal.”

The Game

Suppose you are competing on a game show, and the host presents you with three doors.

Behind one of the doors is a new car, and behind the other two doors are goats. The goal of the game is to choose the door with the car behind it.

After you have picked your door, the host opens a door that you didn’t choose to reveal a goat. Then the host offers you a deal of either keeping your current choice or switching to the remaining unopened door.

The puzzle is as follows: Is there any advantage in changing doors, or do you have the same chance of driving home in the new car, no matter what you do?

The Answer

A common intuition is that switching doors doesn’t affect your chances of winning. We tend to think that since one of them has a car behind it and the other doesn’t, there is a 50 percent chance regardless of which door is picked.

However, this is an incorrect assumption.

In truth, your chances of picking the door with the car doubles if you switch. How can that be? Each door has a ⅓ chance of having the car, so your first choice always has a ⅓ chance of being correct.

When you switch doors, you get the car only if your first choice isn’t right. There was a ⅓ chance that your first door was right and therefore a ⅔ chance it wasn’t right.

Thus, switching gives you a ⅔ chance of getting the car.

Still unconvinced? You’re in good company. This question was originally popularized in the Parade Magazine column, “Ask Marilyn,” where Marilyn vos Savant answered questions that readers sent in.

Marilyn was asked the Monty Hall Problem, and she produced the correct answer.

In response, she received thousands of letters — many of them from Ph.D. statisticians — telling her how wrong she was. Even the great mathematician Paul Erdos remained unconvinced until he was shown a computer simulation of the game.

That’s the best thing about these puzzles: The answers can be tested.

The Evidence

Machine learning algorithms are not as dogmatic as humans. Modern algorithms can sniff out a fraction of a percent increase in productivity in significantly more complex situations, even when it may not seem logical to humans.

If you remain unconvinced, I invite you to search for a simulation online (the UCSD math department and both have versions), or try playing the game yourself with a partner, three paper cups, and a coin.