MARILYN CORNWELL: Aug 23 2022 - The Monty Hall Problem

Monty Hall was the game show host of Let's Make Deal which originated in 1963. He would ask the contestants: which door will you choose? Door number 1, Door Number 2 or Door Number 3. Behind one of these doors is a CAR!!! Behind each of the others is a goat. (I don't remember there being a goat).

The Monty Hall problem is the name of a probability puzzle. It goes like this: 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?

Don't we each have an intuitive response? Of course. This is what happened in 1990. Marilyn Vos Savant's got this question and responded in Parade Magazine. She quoted Steve Selvin's (well-known statistician) answer from 1975.

It was that the contestant should switch to the other door. Under the standard assumptions, 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.

"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."

The problem is a paradox of the veridical type, because the solution is so counterintuitive it can seem absurd but is nevertheless demonstrably true. There are some conditions for this paradox. These are very complicated to me and involve the host behaviour in opening the door randomly and whether the host is required to make the offer. Here are the rules:

The host must always open a door that was not picked by the contestant.
The host must always open a door to reveal a goat and never the car.
The host must always offer the chance to switch between the originally chosen door and the remaining closed door.

This part is very fun in the Wikipedia article HERE with host behaviours described as Monty from Hell, Mind-reading Monty, Angelic Monty, Ignorant Monty, and Deal or No Deal.

In real life Monty Hall wrote to Selvin with this:

"And if you ever get on my show, the rules hold fast for you – no trading boxes after the selection."

Monty Hall was interviewed by the New York Times and Hall demonstrated he understood the problem and gave the reporter a simulation with car keys. He explained how Let's Make a Deal differed from the rules of the puzzle. Go take a look at Hall's explanation - it is fascinating and complicated - just like the puzzle.

I think I would have named him Mighty Monty - he sure knew a lot about game theory and played his guests as they played the Doors.

This is such a pretty picture to me - it was taken at Chanticleer Gardens south of Philadelphia, and features a composition of spring flowers and leaves from the extensive gardens. This was from a visit in 2019, as we haven't been across the border since then. The format comes from Fine Art America and makes a nice way of promoting one's art.