I was watching Brooklyn 99 Season 4 Episode 8 around the 5 minute mark
The problem goes “There are 3 doors behind one of which is a car. You pick a door and the host, who knows where the car is, opens a different door showing nothing behind it. He asks if you want to change your answer.
Apparently the math dictates that you have better chances if you change your decision. Why? 2 doors 50/50 chance, no?
One character (Kevin) says it’s 2/3 if you switch 1/3 if you don’t. What? How? Please help.
In: Mathematics
Caveat to what everyone else is saying: if the problem is not explained very precisely, the true answer is undetermined as there is not enough information to compute.
In particular, it is important that
1. the host knows where the car is, and
2. the host deliberately chooses never to reveal the car
Saying that the host knows where the car is and happened to open nothing in this particular instance is not sufficient information to create the 2/3, 1/3 solution. It is important that the host deliberately avoids the prize, as that is what causes the door he didn’t choose to become special. He has special knowledge, and you can gain partial knowledge from observing his behavior if his behavior depends on his knowledge.
If, instead, the host opens doors randomly and just happened to open nothing this time, then we are not in the true Monty Hall problem, but the “Monty Fall” variant, in which the answer is 1/2, 1/2 as people intuitively suspect.
In a lot of cases, the problem is difficult not because people being given the problem are bad at math, but because the presenters are, and provide an ambiguous scenario which could either be the Monty Hall problem or the Monty Fall problem, and people assume it’s the Monty Fall problem since that’s a reasonable assumption someone would make if not given information that rules it out.
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