Well-known, but don't look up

[Ender]

A random lady has two children, and at least one of them is a boy. What are the chances that the other one is a boy?

[rabbit]

I remember doing this in high school, but I don't remember how.....or what my answer was......dammit.

[iago]

A random lady has two children, and at least one of them is a boy. What are the chances that the other one is a boy?

Well, it depends on genetics. Are some people more pre-disposed to have boys or girls? Could having one boy indicate that mother's pre-disposition, thus making it slightly more likely for her to have another boy?

In any case, I'd say the chances are approximately 50%, unless there's some tricky wording.

(I say approximately because of this)

[Towelie]

Are you asking whats the probability to have two boys in a row? Or just the probability that the next one is a boy? 25% chance that she has two boys in a row, 50% chance that the 2nd one is a boy.

[iago]

Are you asking whats the probability to have two boys in a row? Or just the probability that the next one is a boy? 25% chance that she has two boys in a row, 50% chance that the 2nd one is a boy.

Good point, but I *think* that the first boy is assumed, by the wording of the question.

[abc]

I agree with you iago, but I also was curious if the word "random" put in there, had something to do with tricky wording.

[Ender]

No one has gotten the right answer yet . And yes, we are assuming that a boy is just as likely to be born as a girl.

[iago]

I was just reading this again, and it suddenly reminded me of the three-door paradox. There's a prize behind one door. Pick one. You picked two? Well, let's open three. No prize? That's good. Now, do you want to open the one you picked or the other?

(answer is the other, as long as the other person's selection was random, because the odds are now 50/50 for the other, compared to the 33/66 that you picked the first one with)

So what it comes down to is sort of paradoxical. There are two distinct scenarios, and it does hinge on the word "random" (thanks, Dale!)

If you take a random person with 2 children, there are 4 combinations:
Boy/Boy
Boy/Girl
Girl/Boy
Girl/Girl

For the purposes of this question, we choose a random family. That family has one child who's a boy. That means that it's one of these:
Boy/Boy
Boy/Girl
Girl/Boy

So if one child is a boy, what is the other? 1/3 of the time, it's a boy, and 2/3 of the time, it's a girl.

I don't really understand how it works, the same way I don't really understand how that door problem works, but all I know is, it makes my head hurt trying to think about it.

[topaz~]

A random lady has two children, and at least one of them is a boy. What are the chances that the other one is a boy?

100%?

Never mind, I reread the question again. I thought it was a riddle sort thing where the answer is readily available to you.

[Ender]

Yep. 1/3. Correct, iago!

[Sidoh]

http://en.wikipedia.org/wiki/Monty_Hall_problem

[Ender]

A random lady has two children, and at least one of them is a boy. What are the chances that the other one is a boy?

100%?

Never mind, I reread the question again. I thought it was a riddle sort thing where the answer is readily available to you.

Heh. The wording is tricky, but I made it essentially same as the original wording, taking out the fluff at the start. The word "other" doesn't mean the one who's not a boy. We know that there's one child who is certainly a boy, and one child of uncertain gender, though we don't know which has certain gender and which has uncertain gender. The word "other" refers to the child of uncertain gender.

[Ender]

Yeah, this was a question that a reader asked Marilyn Vos Savant. I think it's called the "Two boys" problem.

[abc]

I was just reading this again, and it suddenly reminded me of the three-door paradox. There's a prize behind one door. Pick one. You picked two? Well, let's open three. No prize? That's good. Now, do you want to open the one you picked or the other?

(answer is the other, as long as the other person's selection was random, because the odds are now 50/50 for the other, compared to the 33/66 that you picked the first one with)

So what it comes down to is sort of paradoxical. There are two distinct scenarios, and it does hinge on the word "random" (thanks, Dale!)

If you take a random person with 2 children, there are 4 combinations:
Boy/Boy
Boy/Girl
Girl/Boy
Girl/Girl

For the purposes of this question, we choose a random family. That family has one child who's a boy. That means that it's one of these:
Boy/Boy
Boy/Girl
Girl/Boy

So if one child is a boy, what is the other? 1/3 of the time, it's a boy, and 2/3 of the time, it's a girl.

I don't really understand how it works, the same way I don't really understand how that door problem works, but all I know is, it makes my head hurt trying to think about it.

Good job, the "random" just stuck out of context to me for some odd reason.

[Ender]

I didn't need to use the word "random," but I just wanted to emphasize that this lady had no proclivity for giving birth to boys over girls.

[iago]

I think the "random" part is important. If you just said, "a lady has two children, one is a son. What are the odds that the other is a son", I think it's more ambiguous. If you went out and found a lady that had a son, then the odds that the other is a son would be, I think, 50%. But if you found a random person, determined that she had one son, and asked what the other might be, that's different.

I think.

[Nate]

Answer is .33

Order isn't important so there are 3 different states, BB, GG, BG
Each has the same probability P, the sum of the probabilities has to equal 1,
so 3P = 1 or P = .33.

[Rule]

I think the "random" part is important. If you just said, "a lady has two children, one is a son. What are the odds that the other is a son", I think it's more ambiguous. If you went out and found a lady that had a son, then the odds that the other is a son would be, I think, 50%. But if you found a random person, determined that she had one son, and asked what the other might be, that's different.

I think.

No.  If you only sought out a lady who had a son and one other child, and you know nothing else about the lady, then there is no difference.

However, if a lady has one son, and she is pregnant, the probability that the child-to-be is a boy is 50%. It seems, superficially, that the situations are identical.  After all, the lady will eventually have one boy, and one other child.  But giving birth shouldn't change the probability that the other child is a boy.

I'll pose this as a new problem .  Explain the difference that accounts for the difference in probability.

[Rule]

Answer is .33

Order isn't important so there are 3 different states, BB, GG, BG
Each has the same probability P, the sum of the probabilities has to equal 1,
so 3P = 1 or P = .33.

? She has one boy.  GG is not a possible state.  You also have to justify why the three actual possible states have equal probability.

[Rule]

Explain the difference that accounts for the difference in probability.

Somehow I think there will be no takers.  lol

[rabbit]

You guys seem to be forgetting that there's also the possibility of hermaphrodites, identical twins, and fraternal twins.