Monty Hall Problem and Bayes’ Rule
https://www.scientificamerican.com/article/the-3-door-monty-hall-problem/
In this article, the author explains what the Monty Hall is and the most common question people ask about it. The question was raised in American television game show and named after its original host, Monty Hall. 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?
Most people intuitively will think that whether to change their original pick does not matter, since there is a 50/50 chance. However, is this really the case? Logically, we know that people would switch if the probability of the car behind No.2 is greater than 1/2. Based on the Bayes’ rule that we discussed in class, the probability of the car behind No.2 is calculated as Pr(car behind 2|goat shown behind 3), which is actually 2/3 as the calculation shows. If this is not convincing, there are simulators available online, so you can try it multiple times or compare the data to other people to see if the overall probability for switching your choices is around 2/3 rather than 1/2.