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
Let’s make a small change to the game — instead of two doors being empty, one has a blue ball inside, and the other has a red ball.
You can see there are four cases:
1. You pick blue 1/3 of the time, monty reveals red. You win by switching
2. You pick red 1/3 of the time, monty reveals blue. You win by switching
3. You pick the car 1/3 of the time
1. monty reveals blue 1/2 of the time. You lose by switching
2. monty reveals red 1/2 of the time. You lose by switching
Crucially 3a + 3b put together are still a total of 1/3, and cases 1 + 2 put together add up to 2/3 probability of winning by switching.
Now, imagine the balls are invisible. You still have the exact same scenarios with the same probabilities, so 2/3 probability you’re in case 1 or 2 and win by switching, you just can’t tell them apart. Invisible balls or no balls at all are fundamentally the same thing.
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