Boy or Girl paradox - Monty Hall spinoff | TigerDroppings.com

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slackster

# Boy or Girl paradox - Monty Hall spinoff

The Monty Hall threads have got me reading about all kinds of statistics and I came across this paradox and thought I would share...

Given: Mr. Smith has exactly two children.

1) If the older child is a boy, what is the probability that both children are boys?

2) If at least one of the children is a boy, what is the probability that both children are boys?

3) If at least one of the children is a boy who was born on a Tuesday, what is the probability that both children are boys?

Assume that the children are not twins nor are either of them an intersex child. Further assume that the chance of being a boy or girl at birth is equal.

I'll hang up and listen.

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re: Boy or Girl paradox - Monty Hall spinoff

50-50. All three

This post was edited on 3/7 at 10:02 pm

TulaneUVA

re: Boy or Girl paradox - Monty Hall spinoff

1/4

eta: giving troll response for troll post

This post was edited on 3/7 at 10:05 pm

slackster

re: Boy or Girl paradox - Monty Hall spinoff

quote:

50-50. All three

castorinho

re: Boy or Girl paradox - Monty Hall spinoff

Lolno

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re: Boy or Girl paradox - Monty Hall spinoff

What are you laughing at.

Okay. The probability is 0.5

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Jet12

re: Boy or Girl paradox - Monty Hall spinoff

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Korkstand

re: Boy or Girl paradox - Monty Hall spinoff

1) 1/2
2) 1/3
3) 13/27 (edit again: shit, right the first time )

This post was edited on 3/7 at 10:26 pm

slackster

re: Boy or Girl paradox - Monty Hall spinoff

quote:

Korkstand

slackster

re: Boy or Girl paradox - Monty Hall spinoff

quote:

giving troll response for troll post

It's a paradox and was specifically labeled as one.

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Hoodoo Man

re: Boy or Girl paradox - Monty Hall spinoff

I hate math that I don't get.

slackster

re: Boy or Girl paradox - Monty Hall spinoff

Conditional probabilities man, they are nuts.

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Peazey

re: Boy or Girl paradox - Monty Hall spinoff

I had never seen this before. I think it's pretty cool.

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tom

re: Boy or Girl paradox - Monty Hall spinoff

There's not really a logical paradox.

The "paradox" in the second problem rests on the fact that there are 2 unspecified children. In solving the problem, the solver is tempted to separate those children into Child A and Child B. This is where the probability calculation fails. There is not Boy-Girl and Girl-Boy in the unspecified system, you can only get that by arbitrarily identifying the children.

The third problem's answer is 1/2 as well. The day of birth is completely irrelevant to the question of sex and should not be part of the calculation at all.

slackster

re: Boy or Girl paradox - Monty Hall spinoff

quote:

The third problem's answer is 1/2 as well. The day of birth is completely irrelevant to the question of sex and should not be part of the calculation at all.

You're incorrect.

quote:

There's not really a logical paradox.

Oh really?

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jeffjo

re: Boy or Girl paradox - Monty Hall spinoff

Tom has the reasons completely wrong, but the right answers. The answer, to each question, is 1/2. Reason: We aren't told why we know (1) The older is a boy, (2) at least one is a boy, or (3) at least one is a boy who was born on a Tuesday, and it matters in the second and third questions. Say, for example, that the family consists of an older boy born on a Tuesday and a younger girl born on a Thursday. For questions (2) and (3) we need to know why we know about the boy, and not the girl. Specifically, if that method would allow us to learn anything else.

If you get 1/3 and 13/27 for the second and third answers, you are assuming it is only possible to know about the boy in such a family. But if it possible to know about the boy or the girl, we can only assume we would know about either 50% of the time, and the answer is 1/2 in each case.

Ironically, the same logic is used to get the correct solution (not just the correct answer, which is usually achieved thru an incorrect solution) in the Monty Hall Problem, just with different numbers. When the logic slackster uses above is applied there, you get that the car is placed behind each door with a probability of 1/3 (compare to each family type in {BB,BG,GB,GG} having a 1/4 probability). One case is eliminated for whichever door is opened (or for GG), leaving 2 (or 3) which are still equally likely. The answer this way is 1/(1+1)=1/2 for the car being behind the remaining door (or 1/(1+1+1)=1/3 for BB).

But if you originally chose the car (or the family is BG or GB), Monty Hall has two choices about what door to open (or there are two facts you could know about the family). You not only have to "remove the cases" where the information you have is false (open door has the car, or family is GG), but also the ones where it is true but you would learn something else (your door has the car but Monty opens a different door, or {BG,GB} and you know about the girl). So the answer is 1/(1+(1/2))=2/3 for the car being behind the remaining door (or 1/(1+(1/2)+(1/2))=1/2 for BB).

The math is slightly more complicated for the Tuesday Boy question, but the result is the same. And again, ironically, my solution was pointed out by Martin Gardner in 1959. He originated questions (1) and (2) in the form presented above, and (3) was posed at a conference named for him. He originally said 1/3 was the answer, but withdrew that answer later that year.