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Monty Hall problem: formula


Kodzikas

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Hello.

Do you know the Monty Hall problem? It states:

"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?".
The contestant should switch to the other door. Under the standard assumptions, contestants who switch have a 2/3 chance of winning the car, while contestants who stick to their choice have only a 1/3 chance.

After doing some research I found that there is a formula that can count the probability of winning if switching doors.
( n-1 )/( n(n-2) ) - that is the formula for the probability of winning if switching doors. The n means total number of doors. (try it when n=3, you get 2/3 of winning if changing doors.)

Overall, the formula tells the probability of winning only when there is 1 door that has the car and the host only opens one door. I didn't find any mathematical formula that could count the probability of winning if there are more doors with cars and the host opens a lot more doors.

So I thought I could create it.

After doing combinatorics and lots of tables. I found the formula:
( x*(n-1) )/( n(n-y-1) ), the x means the number of doors with cars in it and the y means the number of doors the host opens (with the goats). n is the same, mean the number of all doors.
The formula is simplyfied and at first it was a long one with lots of factorials.
Eg.:
We have 5 doors in total, 2 have cars and the host opens 2 doors. At first you choose a door and you have a probability of winning : 2/5, but if you switch doors after the host opens the 2 doors, you have a probability of winning (my formula says) : 8/10=4/5

I made a paper about this (but it's in my native language) and I'm 16 years old.
I want to hear thoughts about my mathematical formula. Has anybody found a paper, that has a formula like that in probability and ect.?
I might have not proven the formula, because it takes a lot to write the proof. But it tells the probability of winning if there is a change (like the host opens the doors).

What do you people think?





Edited by Kodzikas
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I haven't looked in detail yet but I will try to make time. Have you run a simulation to check your answers are correct - you could even do it by copying down on an excel spreadsheet. Just to make sure that you haven't missed something

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