And another puzzle

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, 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?

Switch doors every time. Better chance


PS-have you been reading The Curious incident of the Dog in the Nighttime?

[Viney]

Id be happy with a goat. At least you wont have to pay for it to be on the road...and i'll have a nice lawn!!. Sorry, noy helping much am i.

Originally Posted by Viney

Id be happy with a goat. At least you wont have to pay for it to be on the road...and i'll have a nice lawn!!. Sorry, noy helping much am i.

Happy with a goat?

You must be KIDDING

Taxi !!

[Bluepete]

Bet it can't take off!

Originally Posted by sarah

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, 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?

Two doors left. 50:50 chance. Doesn't matter what you picked before the host opens a door.

Originally Posted by sinbad

Two doors left. 50:50 chance. Doesn't matter what you picked before the host opens a door.

Er, not quite.
The host knows what is behind each of the three doors. If your original choice was the door with the car, then he can pick either of the other two doors to open, which gives you a 50/50 chance. However, if your first choice was a door with a goat (which was a 2/3 probability) then that leaves only one door that he can open - that which has the other goat behind it. Therefore the chances are that the remaining door hides the car.

Does this make sense? I've had a few beers on top of strong painkillers, so might be spouting complete rubbish.

no difference, I'm with sinbad on this.

Assuming you pick the car. The probability of switching is 1/2. You switch, or you don't switch.

We know the host has to pick a goat. That leaves two possible doors. IF you switch, there's a 1/2 probability that the door you switch to has the car (we know it doesn't, but bare with me).

If you don't switch, you have a 1/2 probability of getting the car. We know the 3rd door is a goat because the host picked that one.

OK, so 50:50 there, no matter if you switch or not. There is no advantage in switching.

Let's now assume you pick a goat (but you wouldn't know this yet). The probability of switching is constant, at 1/2. You do, or you don't.

If you do switch, you either started with the car, or you didn't. So if you switch to the other door, there's a 1/2 probability that you get the car. If you don't switch, there's a 1/2 probability that you get the car too.

So to summarise, picking the car first.
You switch to the car, 1/2
You stick with it & have the car, 1/2.
And picking a goat first:
You switch to the car, 1/2
You stick & get the car, 1/2.

Hmm.

The only way the probabilities could be affected, is if you knew what you had already chosen, which would defeat the purpose of the game.

Of course, back in the real world, if the host knows, then you could read their body language etc, and find out. However, this could be countered by the production team knowing, and only telling the host which door to open when the time arose.

Lets look at it like this,

1) The contestant first chooses the door with the car behind it. She is then shown either door A or door B, which reveals a goat. If she changes her choice of doors, she loses. If she stays with her original choice, she wins. 2) The contestant first chooses door A. She is then shown door B, which has a goat behind it. If she switches to the remaining door, she wins the car. Otherwise, she loses.

3) The contestant first chooses door B. She is then is shown door A, which has a goat behind it. If she switches to the remaining door, she wins the car. Otherwise, she loses.

Each of the above three options has a 1/3 probability of occurring, because the contestant is equally likely to begin by choosing any one of the three doors. In two of the above options, the contestant wins the car if she switches doors; in only one of the options does she win if she does not switch doors. When she switches, she wins the car twice (the number of favorable outcomes) out of three possible options (the sample space). Thus the probability of winning the car is 2/3 if she switches doors, which means that she should always switch doors - unless she wants to become a goatherd.