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April 13th, 2017, 12:51 AM   #1
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Cool Help regarding a Modified Monty Hall Problem

Good Day Mates! I'm solving for the probability of winning in a modified Monty Hall Problem where there are 4 doors randomly containing 3 goats and 1 prize. The game mechanics are as follows: 1st a player will randomly choose a door and then the host will open 1 door containing a goat, so that leaves 2 remaining doors that are closed. I want to know what is the probability of winning if the player chooses to switch at any of the 2 remaining doors.

So far, this is my solution:

P(Winning) = 1 - P(Losing)
P(Losing) = losing conditions : (Choosing a the right door the first time) +
(Choosing a door with a goat the first time and
then choosing a door with a goat in the remaining 2 doors)

P(Winning) = 1 - [(1/4) + (1/3 * 1/2)]
P(Winning) = 7/12 or 58.33%

But I did a computer simulation of this problem and empirically it says that the probability of winning is around 36-38%. Can anyone can shed their insights on this problem?

Last edited by skipjack; April 13th, 2017 at 05:56 AM.
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April 13th, 2017, 06:55 AM   #2
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Your mistake is in the probability of choosing a goat the first time.

This should be 3/4 not 1/3.

The result is the win probability = 3/8 = 0.375, which is what you are seeing.

Last edited by skipjack; April 13th, 2017 at 07:33 AM.
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April 13th, 2017, 12:12 PM   #3
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Initially there are 4 doors and one prize. The probability of choosing the prize is 1/4.

Imagine doing this 1000 times. You will choose the door 1000/4= 250 times, one of the 'goats' 750 times. Now, one of the goats is removed. You either choose another door or you don't.

1) You stick with the door you choose. The probability you have the prize is still 1/4= 0.25.

2) You choose another door. In the 250 of the 1000 trials where you had chosen the prize, you will be moving away from the prize to a goat. In the other 750 trials, where you had initially chosen a goat, 1/2 the time, 750/2= 375, you will choose the prize, in the other 375, you will choose a goat.
So, here, there will be 375 times when you choose the prize, 250+ 375= 625 times, you will choose a goat. The probability you choose the prize is 375/1000= 0.375. That is larger than 0.25 so you should switch.
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