The boy/girl probability question (for the 50th time. I know. I'm sorry.)

Great Debates
1

I know that this has gone round many times, but that is exactly why I would like to discuss the interpretation of a certain mathematician’s version of the problem, now that we’ve hashed out everything:

"Consider now some randomly selected family of four which is known to have at least one daughter. Say Myrtle is her name. Given this, what is the conditional probability that Myrtle’s sibling is a brother? Given that Myrtle has a younger sibling, what is the conditional possibility that her sibling is a brother? The answers are, respectively, 2/3 and 1/2.
"In general, there are four equally likely possibilities for a family with two children–BB, BG, GB, GG, where the order of the letters B (boy) and G (girl) indicates birth order. In the first case, the possibility BB is ruled out by assumption, and in two of the three other equally likely possibilities, there is a boy, Myrtle’s brother. In the second case, the possibilities BB and BG are ruled out since Myrtle, a girl, is the older sibling, and in one of the remaining two equally likely possibilities, there is a boy, Myrtle’s brother. In the second case, we know more, accounting for the differing probabilities.

John Allen Paulos, Innumeracy, p. 86

My claim is that Paulos has not formed the first question in such a way that the answer is 2/3. I have no dispute with the second question.

I already affirm (in the well-formed formulation of the question) that the conditional probability that there is exactly one girl and exactly one boy, given that at least one child is a girl, is 2/3. My dispute, of course, concerns Paulos’s phrasing. Either the answer is 1/2, or else the question is ambiguous and cannot be answered unless we add other important assumptions. (Paulos has not told us that in GG families both girls may be named Myrtle, so I assume the opposite – the most natural and obvious assumption.)

Generally, probability questions use names to mark out and identify a unique individual. Identifying a daughter as “Myrtle” or identifying a daughter as “the older sibling” each has have the same effect. As soon as one child is specified, the remaining child has a probability of 1/2 boy and 1/2 girl.

Does anyone think otherwise? Paulos is writing for the public, and he is attempting to combat mathematical ignorance and “illiteracy.” I find it disappointing, then, that he would make this mistake.

The most recent thread on this topic seems to be

http://boards.straightdope.com/sdmb/showthread.php?t=383492

2

The name seems unimportant. He has made no specific statement about the siblings name one way or another and thus it should not change anything.

Am I not understanding your issue?

3

Is it because GG actually has 2 possibilities, one where Myrtle is first and one where Myrtle is second?

4

Think of an equivalent problem, where I flip two coins and tell you that at least one of them came up heads. The conditional probability that the other one is a tail is 2/3. Your assertion regarding the girl’s name is equivalent to saying that knowing that the coin that came up heads is a quarter changes the nature of the problem. How so?

Ah, but here you’re making an additional assumption that isn’t specified in the problem. Of course it’s going to change the outcome.

5

I agree with the OP. The question is phrased all kinds of wrong.

In order to get 2/3rds, I’d think the question would need to read thusly:

***If we randomly selected a family out of population of families who have at least one daughter, what is the conditional probability that the family we select will have a son. ***

Because 1/3 of the population of these families will have two daughters, 1/3 will have an older daughter and a younger son, and 1/3 will have an older son and a younger daughter, the answer is 2/3rds.

The answer becomes 1/2 when the question becomes:

Given that Family X has a daughter, what’s the probability that Family X also has a son?

The OP in the linked thread was also poorly phrased because the family is given to us upfront.

6

To elaborate further, the 2/3rds answer only works if you are determining the odds of a certain outcome associated with random selection. Where the question in the OP messes up is where it asks about Myrtle specifically. Insteading of asking us to consider the odds that a family of 4 will have a girl and boy on the condition that this family as at least one girl, the introduction of Myrtle changes the question to “what’s the probability that Myrtle, who happens to belong to a randomly selected family of 4 that has at least one daughter, has a brother?” Those are two different questions.

Unless all the families that make up the population from which we are selecting from have a daughter named Myrtle, the author is introducing a factor that inteferes with what should be a fairly straightforward word problem.

7

The sample space is BB, BG, GB, GG, with all outcomes equally likely. The probability that there’s a daughter is 3/4, and the probability that there’s a daughter and a son is 1/2. The conditional probabilty that the family has a son given that they have a daughter is 1/2 divided by 3/4, or 2/3.

8

Here’s your problem right away. Identifying a daughter as “Myrtle” tells us nothing about whether the family she’s part of has the BG, GB or GG pattern. Identifying a daughter as the elder sibling rules out BG along with BB. Hope this helps.

Unless we know that there is something screwy about the distribution of families containing a daughter named Myrtle as opposed to families containing a daughter of indeterminate name, the factor isn’t interfering with a darned thing.

9

That’s not the correct answer if the family is given to you upfront.

Given that the Smith Family has two children, one of whom is a girl, what’s the probability that the other child is male?

You only have two possible outcomes. The second child is either a girl or is a boy.

The answer is 2/3 only when you’re randomly drawing a family out of a hat and you’re being asked to predict the odds of selecting a family that has a boy and a girl. But once you’ve made that selection and the question becomes “what’s the probability that this particular daughter has a brother?” the answer is 1/2.

10

If you’re going to treat one boy and one girl as one outcome, you can’t assume that it’s as likely as two boys or two girls, because that’s wrong.

If you agree that the gender of a child is independent of the gender of any previous children, and that boys and girls are equally likely*, then the number of girls in a family with two children is a random variable with a binomial(2, 1/2) distribution. The probability that it’s equal to 1 is 1/2.

11

And there’s your problem right there, you with the face. You have asserted “You have only two possible outcomes” and you have jumped to the conclusion “Therefore, they are equally likely”; begging the question, unless I’m very much mistaken.

12

Put this in the realm of reality if you disagree with me. Given that ywtf has a fraternal twin (which is a fact), what’s the odds that her twin is her brother?

Do you disagree that there is only two possibililties to choose from? Either my twin is male or female. Therefore, there’s a 1/2 chance that my twin is my brother.

Given that the Smith Family has two children, one of whom is a girl, what’s the probability that the second child is a boy?

I only count two permutations. The second kid is either a boy or a girl. How many do you count?

13

you with the face’s problem is that he is, quite correctly, arguing that a different question to the question in the OP would have a different answer. He fails to understand the question in the OP.

you with the face is saying - pick a random family from all the families with 2 children. If one child is a girl what is the chance the other is a girl?

The problem posed in the OP is, pick a random family from all the families with 2 children except those that have 2 boys. What is the chance that they have 2 girls?

14

But I haven’t done so. They are two separate outcomes, but you are only being asked about one. If we know that the family has one girl, then the question being asked is basically is no different than asking what’s the chances that a baby will be a boy. It’s 50:50.

Right. Which is why the answer is 1/2 if you’re being asked about a specific family. If you are being asked about a random family in a population of families, the answer is 2/3rds.

15

Hey-ho. When in doubt, change the hypothetical.
As to the Smith Family, I count only two possibilities for the gender of the other child, but I count three possible permutations for their family, given that “two boys” has been ruled out by the terms of the question:

An older girl, a younger boy
An older boy, a younger girl
Two girls

Do you believe that these three permutations are not equally likely, if we accept that the gender ratio is close enough to 50-50 for our present purposes?

(Aside to don’t ask: 'face is female.)

16

That’s OK. It’s like technical writing or referring to deities - if I don’t know next time I’ll refer to him as her.

17

That’s not what I’m saying. The correct way to phrase the question is the OP is “On the condition that a randomly selected family of 4 will have at least one daughter, what’s the chances of selecting a family that has a boy?”

The answer to that is 1/3. Wrong answer.

18

What does it matter about the birth order if the question doesn’t require that information? Remember, this is the question I asked you:

Given that the Smith Family has 2 children, one of whom is a girl, what’s the probability that the second child is a boy? All you know is that one of the kids is a girl. If you want to count out all the possible combinations, though, there would actually be four, not 3 like you say. Let’s call the girl that we know about Myrtle.

  1. Older Myrtle, younger brother
  2. Older boy, younger Myrtle
  3. Older Myrtle, younger sister
  4. Older sister, younger Myrtle

In the question I asked, whether or not Myrtle is older or younger is irrelevant, because a brother is a brother regardless of birth order. That means that there only 2 possible outcomes that we need to worry about. So the answer is 1/2.

19

In answer to your fraternal-twins question, you with the face - it doesn’t make any difference that they’re twins. You still have two independently-fertilised ova - call them Ovum 1 and Ovum 2. Neglecting circumstantial bias that might be applicable to gender-selection in twin pregnancies*, you have the following possibilities:

Ovum 1 becomes a male zygote, Ovum 2 becomes a male zygote
Ovum 1 becomes a male zygote, Ovum 2 becomes a female zygote
Ovum 1 becomes a female zygote, Ovum 1 becomes a male zygote
Ovum 1 becomes a female zygote, Ovum 2 becomes a female zygote

You still have four cases, all equally likely. You still have the same gender distribution in fraternal twin births: 25% all-male, 25% all-female, 50% mixed. I know you’re not the product of an all-male birth so you’re one of the remaining 75% of cases, making it 2-1 on that your twin is a boy.
*F’rinstance, if it’s the case that the fast-swimming Y sperm get there first but die off quicker, or one kind of sperm gets favoured by environmental conditions in the vagina, the sperm may be roughly sorted to the extent that conception favours one sex over the other at the time it occurs, and it may in fact be the case that single-sex fraternal twins are more common than mixed-sex. But in the absence of data, I’ll leave this as a footnote.

20

Once you apply the logic you used here to the Smith Family, you’ll see that the answer is the same.