The Monty Hall Problem
“The Rule of Five estimates the median (middle point) of a population. Half of the population is above a certain measure, half is below. There is a 93.75% chance that the median of a population is between the smallest and largest values in any random sample of five from that population. It might seem impossible to be 93.75% certain about anything based on a random sample of just five, but it works.
‘This explanation comes from Marilyn Vos Savant, the Parade columnist who initially popularised the Monty Hall Problem. Basically, you need to consider how much information you have about the doors, because the problem delivers different amounts of information for each door, and if you maximise the info then you maximise your chance of picking the door with the car.
From the beginning, the contestant only has a 1 in 3 chance of picking the right door. When the host reveals one of the goats, it doesn't actually change the odds that the contestant initially picked correctly — their first choice remains a 1 in 3 shot.
However, after one of the goats is revealed, the contestant has a lot more information than the initial 1 in 3 random choice.
The contestant now effectively has a choice between the "one-of-three" door they chose originally or "two-of-three" remaining doors they initially knew nothing about. That is a choice between sticking with your 1/3 chance or going with a 2/3 chance. And, as you know that the car is not behind one of the 2/3 doors, then the car must be more likely to be behind that other door.
So you must switch to get a 2/3 chance of selecting the car!’
Courtesy of Jim Edwards at Business Insider