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Bayesian Statistics Riddle

label Statistics
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There is a car behind one of three doors.  You make a guess that the first door has the car.  After stating your guess, one of the two remaining doors open to reveal nothing.  Does your chance of getting a car increase if you decide to switch doors?

Oct 22nd, 2017

Suppose we have n doors, with a car behind 1 of them. The probability of choosing the door with the car behind it on your first pick, is 1n.

then opens k doors, where 0≤k≤n−2 (he has to leave your original door and at least one other door closed).

The probability of picking the car if you choose a different door, is the chance of not having picked the car, which is n−1n, times the probability of picking it now, which is 1n−k−1. This gives us a total probability of

n−1n1n−k−1=1nn−1n−k−11n

 opens no doors, k=0 and that reduces to 1n.

For all k>0, n−1n−k−1>1 and so the probabilty of picking the car on your second guess is greater than 1n.

If k is at its maximum value of n−2, the probability of picking a car after switching becomes

1nn−1n−(n−2)−1=1nn−11=n−1n

For n=3, this is the solution to this riddle


Feb 13th, 2015

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