The BEST way to explain Monty Hall Paradox!!

Factual Questions
1

Ok, So I’m a little behind the times, but I just finished reading through the Cecil Archive on the Monty Hall Paradox (Search to refresh your memory). There was a long debate whether or not Cecil was right, and at the end alot of readers weren’t satisfied with the explanation. I love Cecil (in that way) but his explanation for this was pretty lame. I got the feeling even he didn’t “get it” even though he was right. Anyway, I got to thinking about it and drew a diagram below that makes it absolutely clear to anyone (I think). Tell me if you agree.

Ok, In the following example, I start with the prize behind door number three (marked by a ! following the X). It doesnt matter which door it starts behind so I just picked any. Then, on the y axis, I listed the three possible situations after you have made your inital choose (since you have the option of three doors). Now, knowing that Monty will not open the door with the prize, and will not open the door you have chosen, watch how this new info changes the odds.

       Door1         Door2          Door3

Poss1 X* O X!

Poss2 O X* X!

Poss3 O O X!*

In Poss1 you choose door number 1 as marked by the * after the X. Monty only has the option of opening door 2 (because you have chosen door1 and the prize is behind door3. In this situation, switching will DEFINATELY win you the prize.

In Poss2, you choose door2, Monty MUST open door1, and once again switching DEFINATELY wins you the prize.

In Poss3, switching will always make you lose.

If, after the new information, you switch, you will win 2 in the 3 situations. Hence the odds of winning = 2/3.

Tada! Ok someone give me kudos for this cause I crave validation.

Having said that, I haven’t figured out something about the two children question (refer to same article). While the odds statiscally do say there is a 2/3 chance of the next child being male, clearly, (as far as I know), betting on this in real life will be a losing bet. In the Monty Hall bet you will actually win money (Yes play the game with friends and play the contestant). Can anyone explain how the children works statistically but doesnt seem to work in real real life?

2

Even if you don’t want to provide a link, I think this should be in Comments on Cecil’s Columns. Maybe I’m wrong, though.

3

KidCharlemagne wrote:

It does work in real life. As a matter of fact, doing it in real life, or at least running through a simulation, will make it intuitively apparent why it works.

4

Once again sorry about multiple posts, me newbie.

While I see that it works if you make out slips of paper and draw them from a hat, I can’t believe that if I have a daughter, I have a 2/3 chance of having a son next. While obviously the two are different, I don’t see how because there is no new information like there is in the MH paradox.

5

You’re falling into the same logical trap everyone else seems to. You’re assuming that this 'paradox says something about the odds of a particular child’s sex. It doesn’t, not really.

You aren’t told that the family had a daughter first and then asked to guess what they will have for their second child. Instead, you know that they already have two children, one of them is a girl, and that the odds of male or female birth are 50/50. The difference is that you can make a matrix of the births, and the female can be in either the older or younger position, or both, or neither. This gives you four possible combinations of sexes. Then, like the Monty Hall example, you can drop one case (the one with two boys). That leaves you with 3 options, 2 of which have boys.

By the way, this really should be in CCC. Here’s the column:
On “Let’s Make a Deal,” you pick Door #1. Monty opens Door #2–no prize. Do you stay with Door #1 or switch to #3?
and here are some previous CCC threads on it:
http://boards.straightdope.com/sdmb/showthread.php?threadid=2095
http://boards.straightdope.com/sdmb/showthread.php?threadid=90
http://boards.straightdope.com/sdmb/showthread.php?threadid=629

6

I know that when you do a matrix of the probabilities it works out that way, but it still doesnt seem the same as monty hall. In other words, I see the children paradox as a trick question. In other words people are duped into thinking of them as individuals vs. a set. And there is no intelligent being who is changing the odds through his actions. I believe it is the same, its funny how I just cant apply the MH puzzle to this one.

7

I’m closing this and the other copy of this thread here in GQ. Since it is, after all, a comment on Cecil’s column, I suggest that we continue the discussion there.