196 replies to this topic

[Riddari]

My explanation of the logic is a bit different and possibly easier to understand.

Spoiler for hint

Let's say the scenario were a little different.
After Jennifer picks Door #2, Monty Barker does not open Door #1. Instead, he decides to be generous and offer Jennifer to trade what is behind Door #2 for what is behind both Door #1 and Door #3.
Assuming that Monty is not intentionally trying to entice Jennifer away from Door #2, it obviously makes sense to trade.
The odds of the Jaguar XJS being behind Door #2 are one in three.
The odds of the Jaguar XJS being behind either Door #1 or Door #3 are two in three.
Jennifer knows she will get at least one goat if she trades, but she also doubles her chances of getting the car.

Now, back in the real scenario, it does not matter that Monty shows Jennifer the Goat behind Door #1. We already knew there was a goat behind at least one of the remaining doors. Switching still doubles her chances of getting the car. The goat behind Door #1 has just been discarded before she wins it, instead of afterwards.

Spoiler for hint

If the above still does not convince you, then maybe this one will. This is the method my friend used to convince me after I argued with him for quite a while about it.

Change the scenario from three doors to a hundred boxes.
Now, there are one hundred boxes to pick from. One has a key to the fabulous Jaguar XJS, the other ninety-nine are actually boxes of Kleenex.
After Jennifer picks box #54, Monty opens all of the other boxes, except #43, to show that they are all tissues.
Finally, he offers Jennifer a chance to trade her box for #43.
Should she make the trade or keep the box she has.
It should be obvious at this extreme that there is still only a one percent chance of box #54 having the key, but a ninety-nine percent chance that #43 has the key.

[aZameGa]

Riddari,

That second spoiler is amazing. makes it pretty clear for some people... like roommates

peace...

[bonanova]

edit: ha! now I've read Riddari's first spoiler. This post adds nothing to what he said.

The way I finally understood the answer is this:

Spoiler for hint

if you stick with your first selection your chances of winning are 1 out of 3. If you switch, you cover the other two possibilities and your chances have to be 2 out of 3.

That one of the two doors you get by switching has a goat is a red herring. you knew that already. Doesn't change your chances.

Riddari's explanation #2 is compelling, also - it's the same situation, magnified, and makes intuition favor the correct answer. It took me an hour to believe the 1 of 3 case.

[cpotting]

If the above still does not convince you, then maybe this one will. This is the method my friend used to convince me after I argued with him for quite a while about it.

Change the scenario from three doors to a hundred boxes.
Now, there are one hundred boxes to pick from. One has a key to the fabulous Jaguar XJS, the other ninety-nine are actually boxes of Kleenex.
After Jennifer picks box #54, Monty opens all of the other boxes, except #43, to show that they are all tissues.
Finally, he offers Jennifer a chance to trade her box for #43.
Should she make the trade or keep the box she has.
It should be obvious at this extreme that there is still only a one percent chance of box #54 having the key, but a ninety-nine percent chance that #43 has the key.

The concept is correct, but I think the reasoning is stated imprecisely. I would word it as follows: there is a 1:100 chance she picked the right box (#54) and, conversely, a 99:100 chance that she didn't. Obviously, she should switch - there is a 99% chance that some box other than the one she picked has the car - but which one? Once 98 of the 99 boxes are shown to be the wrong ones and only box #43 remains, it becomes obvious which she one she should switch to.
I'm not disagreeing with you Riddan, I just changing the wording a bit - it seems clearer to me, but, as they say, to each their own.

The same logic applies to the original problem - just replace 100 with 3, 99 with 2 and 98 with 1.

[Riddari]

I agree that your wording makes more sense, cpotting. That is what I was trying to get at.

[Skini]

I disagree with carrying the odds from the first choice to the second. I think the choices represent two separate independent events. In the first you have a 1:3 chance of picking the car. The second event is a choice between two doors with one prize. I say 50 / 50.

[Martini]

I disagree with carrying the odds from the first choice to the second. I think the choices represent two separate independent events.

You pick a door. If you pick a door with a goat (most likely), the host will show you which of the other door's also have the goat behind it. That makes switching to the other door the best bet.

For instance, if you pick the door with goat #1 behind it, the host will show you the door that has goat #2 behind it and by switching you will win a car.

If you pick the door with goat #2 behind it, the host will show you the door that has goat #1 behind it and by switching you will win a car.

If you pick the door with the car behind it, the host will show you a door that has one of the goats behind it and by switching you will win the other goat.

By switching doors when offered, you have a 2 out of 3 chance of winning the car.

Play with this a little and you'll see how it works.

[Jkyle1980]

The two events are completely independent of each other. You are all looking at this the wrong way (except skini). It is already decided that no matter what your first choice is, you will be shown a goat. That means that there is no probability. It will happen. It is just a little trick to add to the suspense. It will always happen. So since we are dealing with probability and possibility you can rule it out. It's 1. It will happen. So you are essentially looking at a really snazzy toss up.

Someone tells you that they are thinking of a letter, A, B, or C. You guess B. He or she says, "Well, it's definitely not A." So that doesn't mean anything. It means it is either B or C. You're odds are now 50/50. It is a probability between two choices.

[schmod54]

The two events are completely independent of each other. You are all looking at this the wrong way (except skini). It is already decided that no matter what your first choice is, you will be shown a goat. That means that there is no probability. It will happen. It is just a little trick to add to the suspense. It will always happen. So since we are dealing with probability and possibility you can rule it out. It's 1. It will happen. So you are essentially looking at a really snazzy toss up.

Someone tells you that they are thinking of a letter, A, B, or C. You guess B. He or she says, "Well, it's definitely not A." So that doesn't mean anything. It means it is either B or C. You're odds are now 50/50. It is a probability between two choices.

The second choice is definitely not independent, because the options presented to you in the second choice are dependent upon your first choice. And having two choices does not mean that there is equal probability between the two.

The following is a probability tree that shows all possible outcomes if you initially choose door B. C represents a car. G represents a goat. X represents a door eliminated by the host.

1/3    1/3    1/3 CGG    GCG    GGC  |     / \     | 1/3  1/6 1/6  1/3 CGX  XCG GCX  XGC

The total probability that the car is behind door B in the second choice is 1/3. The total probability that the car is behind the "other door" (which is A or C) is 2/3.

[Jkyle1980]

Look at the final scenario not the initial ones. You have two doors. One has a goat behind it; one has a car. It's 50/50. The rest is just smoke and mirrors.