Steven Strogatz
Why do so many people get Monty Hall problem (also known the three doors problem) wrong? Steven Tijms explores both the math and the psychology in this illuminating article: https://chance.amstat.org/2022/11/monty-hall
Monty Hall and the ‘Leibniz Illusion’ | CHANCE

Steven Tijms Seeing Is Believing Throughout the history of mathematics, quite a few mathematical problems have achieved celebrity status outside the circle of mathematicians. Famous problems such as squaring the circle or proving Fermat’s last theorem have intrigued thousands of people over the centuries. But almost no mathematical problem has been as fiercely and widely […]

Dec 24, 2022 at 4:14pmWeb
Show threadJeroen van DorpDec 24, 2022
@stevenstrogatz I always wondered what “probability” means when you only have one observation. I always thought probability to be defined as the quotient of the number of times you observe an outcome and a large number of observations. The probability of succes actually here doubles. De Vos Savant was right there, but was she right with her conclusion that changing doors helps? What does probability mean with only one observation? Do you get 2/3 of the prize? Or only for 2/3 of the time?
Show threadryk047Dec 24, 2022
@jeroenvandorp @stevenstrogatz It's more about misdirection than probability. If Monty didn't open a door but instead offered you whatever's behind the two doors you didn't pick, you'd switch because you'd believe you're going from 1/3 -> 2/3 odds. The misdirection, opening a door with no car behind it, changes nothing: the doors you didn't pick always have one that's without a car. But it does misdirect you and make it seem you're in a 1/2:1/2 world.
Show threadJeroen van DorpDec 24, 2022
@ryk047 @stevenstrogatz Absolutely true. But for me the point is that De Vos Savant claimed that because the probability doubles from 1/3 to 2/3 you should switch doors. The first observation is true, no doubt, but I never understood why the second conclusion should be true.
Show threadryk047Dec 24, 2022
@jeroenvandorp @stevenstrogatz I think her explanation carries the misdirection over. If she'd said the probability of the door Monty opened, zero, plus the probability of the one he offered you to switch to, it might have helped. Still, you start with the notion the probabilities are 1/3:1/3:1/3, then are fooled by the reveal of the open door to believe they change. De Vos Savant's explanation is wrong in the sense that if you switch you're getting what's behind two doors.
Show threadrrogersDec 24, 2022
@stevenstrogatz I didn't believe it until I sat down (I was annoyed) and ran the math. Self-quote "I guess I don't know probability as instinctively as I thought."
Show threadJames AnnanDec 24, 2022
@stevenstrogatz well part of the problem is that it’s routinely presented in an ambiguous manner as a one-off set of events that happened rather than a game that is being played according to predefined rules.
Show threadDave NearyDec 25, 2022
@stevenstrogatz It's pretty obvious to me. People psychologically see picking between 2 options as a 50/50 chance, but they don't consider their first choice as dependent (only 1/3 chance of choosing prize on first choice, so switch = 2/3).
Show threadAndrew Jon ThomsonDec 25, 2022
@stevenstrogatz I read the whole thing and I still don’t understand why the two options are not equally possible. The article is primarily concerned with why people have this illusion but it did not sufficiently explain to my dense brain why the correct solution is correct. Now I need to find an article explaining why.
Show threadAngus MackenzieJan 8, 2023

@HelloAndrew @stevenstrogatz Two thirds of the time on your first choice, you pick a goat. You are always shown a goat - two thirds of the time it is the other goat.

One time in three the host has a choice of two goats and you are about to move away from the car. The other two thirds, you are moving from one of the goats to the car, so overall you should change.

Show threadAndrew Jon ThomsonJan 8, 2023
@anxiousmac @stevenstrogatz I feel like the more I think about this the stupider I get
Show threadbetonglenJan 8, 2023

@stevenstrogatz

thank you for sharing. 🥰🖖🏽

reading now but would love to have this article and topics presented as mini tutorial series. I'm going to test this out myself #ThreeDoors #statistics #probability #Randomness