- 05 Jan '07 13:14You're on a game show in front of three doors, behind two of them there are goats and behind the other one is a car. You must pick a door and win whatever is behind. After you pick a door (say #1) the host opens another door (say #3) and shows a goat, he then offers you the chance to switch doors (to #2 instead of #1)... should you switch? (demonstrate why or why not)

If you want to get technical here are the rules for the game show:

- Always two goats and 1 car.

- Host MUST always open a door containing a goat after the player has picked a door.

- If the player initially chose the car door, the door uncovered by the host is random. - 05 Jan '07 14:30It behooves you to switch. Your initial chances of selecting a door with a goat are 2 in 3.. you are twice as likely to have a goat than the car. Then, the host shows you the other goat. Therefore, switching brings you to the car. If you initially chose the door with the car (a 1 in 3 chance) you lose by switching. Changing doors gives you a 66.6% chance of winning the car.
- 05 Jan '07 14:41You MUST switch every time.

your first choice is a 3:1 ratio.

your "switched" choice is 2:1 ratio.

This is prooved if you think of 100 doors , 99 containing goats and 1 containing a car.

If you choose 1 door and the other 98 doors are opened to show goats. A switch is in your favour (favor) . you win 99 times and lose one time. - 06 Jan '07 00:28 / 2 edits

Ah ha a nice counter intuitive problem that illustrates the hard wierd non logical way in which our minds work (or not work as is more reasonable to say).*Originally posted by trad***You're on a game show in front of three doors, behind two of them there are goats and behind the other one is a car. You must pick a door and win whatever is behind. After you pick a door (say #1) the host opens another door (say #3) and shows a goat, he then offers you the chance to switch doors (to #2 instead of #1)... should you switch? (demonstrate why or w ...[text shortened]... door.**

- If the player initially chose the car door, the door uncovered by the host is random.

The first choice gives us a 1in 3 probability of choosing the car. Then through being presented with the option of switching we reason intuitivly that we now have a 50:50 probability in choosing the car.

However the true probability(at least in a mathematical definition of the system) is 2 in 3.

We have simply choosen 2 out of the 3 boxes.

So we should always switch.

You have all fallen into the intuitive mistake when visualising the problem. . . . .In saying that the final probability is 50:50 by breaking the problem into two distinct choices. - 06 Jan '07 14:03Okay I've read this before and I understand the calculations above. But still it puzzles me.

Let's say the player picks door 1. Then door 3 with a goat is opened. So after that he has a higher chance if he switches to door 2. But just before he makes a decision, he gets replaced by another candidate who can choose from the remaining doors 1 and 2. So the new player also has a higher chance if he picks door 2? Or only if he has witnessed what happened before? Witnessing or not, won't change what is behind door 2. - 06 Jan '07 16:48

A new contestant, with no knowledge of what transpired, is choosing between two doors, one with a goat and one with a car. His/her chance is 50%. If the new contestant witnessed what transpired, his/her chance to win the car by switching doors is 66.6% -- the same as the original contestant.*Originally posted by crazyblue***Okay I've read this before and I understand the calculations above. But still it puzzles me.**

Let's say the player picks door 1. Then door 3 with a goat is opened. So after that he has a higher chance if he switches to door 2. But just before he makes a decision, he gets replaced by another candidate who can choose from the remaining doors 1 and 2. So th ...[text shortened]... he has witnessed what happened before? Witnessing or not, won't change what is behind door 2. - 07 Jan '07 01:57 / 1 editSay you pick door #1

If car is behind:

1 - switch and you lose

2 - switch and you win (because host would open #3)

3 - switch and you win (because host would open #2)

The key is that the host knows where the car is, and always opens a losing door.

If you stay with # 1, however:

If car is behind:

1- stay and you win

2 - stay and you lose

3 - stay and you lose

If you switch, you win 2 out of 3 times.

If you stay with door #1, you only win 1 out of 3 times.

So yes, you should switch.