I’ve watched a few videos but I can’t fully understand it
In: Mathematics
If I put you in a room with no windows and asked you the probability that it will rain today, what would you say? Now suppose I let you look outside. Would seeing the sky change your assessment of the probability of rain? If you said yes, then we agree that new information can change previous probabilities. And that’s really what the Monty Hall problem is about: Probabilities are not fixed. They change as the available information changes.
Let’s say that, instead of 3 doors, there are 100. You have a 1% chance of picking the car and a 99% of picking a goat. Now, Monty opens 98 of the doors to reveal goats, so all that’s left unopened is your door and some other door. Now that you have 98 more pieces of information, do you think it’s more likely that the car is behind the door you originally picked or the one that Monty (seemingly at random) decided not to open? You might think it’s 50:50 since there’s only 2 doors left, but is it really? Given the rules of the game and the fact that Monty neglected to open *this one particular* door?
When people think that the new probabilities should be 50:50, it’s because they are neglecting that the previous choice ever happened, as though there were only ever the two final doors. That is a very important piece of information to neglect. The problem with only 3 doors follows the same logic, just on a smaller scale.
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