Saturday, February 4, 2023


A few days ago I posted a paradox known as the Ship of Theseus, which asks how much of a ship has to be replaced before it stops being the original ship.

Here is an example of another fascinating paradox. . . .

It is known as The Monty Hall Problem for being based on the American television game show Let's Make a Deal, the original host of which was Monty Hall.

The problem was originally posed (and solved) in a letter by Steve Selvin to the American Statistician in 1975. It became famous as a question from reader Craig F. Whitaker's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Do you stay with Door 1 or switch to Door 2? Does it make any difference? Do the odds improve, worsen or stay the same by switching?

Surprisingly, the odds aren’t 50-50.

The switching strategy has a 2/3 probability of winning the car, while the strategy that remains with the initial choice has only a 1/3 probability.

When Monty offers the switch after revealing that one of the doors does not have the car, the majority of people assume that both remaining doors are equally like to have the prize, that each of the 2 doors has a 50/50 chance. Because there is no perceived reason to change, most people stick with their initial choice.

However, If you switch doors, you double your probability of winning.

Flipping a coin will have a 1 in 2 chance each time of heads or tails. That is because the flipping result is random each time and the probabilities do not change. The Monty Hall problem does not satisfy either requirement.

The only random portion of the process is the first choice. When you pick one of the three doors, you truly have a 1 in 3 probability of picking the correct door.

The process stops being random when Monty Hall uses his insider knowledge about the prize’s location. When it’s time for him to open a door, there are two doors he can open However, Monty doesn’t want to reveal the prize. Monty knowingly opens only a door that does not contain the prize. The end result is that the door he doesn’t show you, and lets you switch to, has a higher probability of containing the prize. That’s how the process is neither random nor has constant probabilities.

If you pick the incorrect door by random chance, the prize is behind one of the other two doors.

Monty knows the prize location. He opens the only door available to him that does not have the prize.

By the process of elimination, the prize must be behind the door that he does not open.

The Monty Hall problem is a statistical, rather than an optical, illusion.

Our mental assumptions for solving the problem do not match the actual process. Our mental assumptions were based on independent, random events. However, Monty knows the prize location and uses this knowledge to affect the outcomes in a non-random fashion. Once you understand how Monty uses his knowledge to pick a door, the results make sense.

If you are still sceptical, non-plussed or confused, you are not alone.

Many readers of vos Savant's column refused to believe switching is beneficial and rejected her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them calling vos Savant wrong. Even when given explanations, simulations, and formal mathematical proofs, many people still did not accept that switching is the best strategy. Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation demonstrating vos Savant's predicted result.


By the way, I love the scene in Labyrinth where Sarah quizzes the guards as to which is lying and which is telling the truth . . .

Sarah: Would he tell me that this door leads to the castle?
Guard: [Whispers with his counterparts] Yes?
Sarah: So... the other door leads to the castle and this one leads to certain death.
Guard: [All the guards Oooh] But he could be telling the truth!
Sarah: But then you wouldn't be. So if he told me that this door leads to the castle, the answer you should give me would be 'No'Guard: But I could be telling the truth!
Sarah: But then he would be lying. So then if he said this door led to the castle, I'd know the answer would still be 'No'
Guard: Is-is that right?
Guard: [snickers] I don't know! I've never understood it!


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