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human
Monty Hall Problem (10th Grader With No Understanding of Statistics)

So in the Monty Hall problem, switching should increase your odds of winning, but what I don’t understand is how that really works. The way it was explained to me is that you have a 1/3 chance the first attempt, and once one door is eliminated switching gives you a 1/2 chance. Assuming that the person explaining it to me explained it correctly, shouldn’t the 33% chance become a 50% chance of you getting it right by not switching since the prize itself doesn’t move and eliminating one door would basically be the equivalent of there being two doors to begin with? Thank you!
gpt
> shouldn’t the 33% chance become a 50% chance of you getting it right by not switching since the prize itself doesn’t move and eliminating one door would basically be the equivalent of there being two doors to begin with?

Kind of, but not quite.

The prize isn't moving, and you are getting down to just two doors, but an important consideration is that you are not re-picking your door from scratch. Your selection of an initial door affects how the host will reveal doors. The host will reveal a door that you did not pick, but will NOT reveal the prize, right?

If you picked the winning door (1/3rd chance), then the host can open either of the other doors, it doesn't matter, since they are both loosing doors.

But if you picking a losing door, then the host has two options: Open the other losing door, or open the winning door. They are not going to open the winning door, so their choice is constrained to the losing door. This means that the other door MUST be the winning door, so that if you switch, you win.

Now, just seeing a losing door being opened by the host doesn't let us know whether switching will make us win or not. It's impossible to know. But we can go back to that original probability: We only had a 1/3 chance to pick the winning door off the bat, which means there is a 2/3 chance we picked a losing door, and in that 2/3 chance, the host opens the *other* losing door, which means if we switch we *have* to win. Hence, switching gets us the 2/3 chance to win. It's not a certainty, but it tilts the chances in our favor.