# The Game Show Problem Explained

The Game Show Host Problem

The problem: 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 #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?

Explanation: The reason is that the first door has a 1/3 chance of winning whereas the second one has a higher chance of 2/3.

[Mostly people consider it this way: when the game show host opens one of the losing doors the chances of winning with each of the other doors is ½. Neither of the other 2 doors has a higher chance of winning, so you have the same odds of winning the car with door#1 or #2.]

Fallacy:

Firstly one should clearly understand that just the opening of a losing door by the host does not increase the chances for door#1 from 1/3 to ½.

So, one way of explaining could be taking all the exhaustive cases of the situation in account. Check out this table:

The following table comprises of the 6 exhaustive cases. The initially chosen door is door #1 and the game show host always opens a losing door.

Game#

Door1

Door2

Door3

Result

1

Car

Goat

Goat

Switch and lose

2

Goat

Car

Goat

Switch and win

3

Goat

Goat

Car

Switch and win

4

Car

Goat

Goat

Stay and win

5

Goat

Car

Goat

Stay and lose

6

Goat

Goat

Car

Stay and lose

So, from the above table it can be seen that on switching one wins 2/3rd of the time (66.7%) whereas on staying one wins only 1/3rd of the time (33.3%).

Now that’s for convincing the stubborn..:-)

Final Explanation:

So, what about the “1/2 chance of winning” concept?

Well, if the game show host would have asked someone completely unaware of door#3 (i.e. the door that the host opened) then using simple probability certainly the chances of winning are 50% for both the doors. What has made the difference is the advantage you have in the form of the knowledge of Door#3.This is the advantage: if the prize is behind door#2 the host would open #3 and if the prize is behind #3 the host would open #2 so if you switch you win if the prize is behind #2 or #3 but, if you stay then you win only if the prize is behind #1.

I hope by now you must have understood that math is not only for studying to get high scores in school…….math can actually be used in ways you would have never imagined!!!!

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