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One Girl - One Boy

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Ok, so Teanchi and Beanchi are a married couple (dont ask me whose he and whose she)!

They have two kids, one of them is a girl, what is the probability that the other kid is also a girl.

Assume safely that the porbability of each gender is 1/2.

Ofcourse its not 1/2 else would make it a lousy puzzle...

Ans: 1/3

This is a famous question in understanding conditional probability, which simply means that given some information you might be able to get a better estimate.

The following are possible combinations of two children that form a sample space in any earthly family:

Girl - Girl

Girl - Boy

Boy - Girl

Boy - Boy

Since we know one of the children is a girl, we will drop the Boy-Boy possibility from the sample space.

This leaves only three possibilities, one of which is two girls. Hence the probability is 1/3

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Hi. I have just read the question and then read some of the forum explainations
(only some of them). Many people are getting themselves tangled up.

A couple have two kids, one of them is a girl.
Assume safely that the probability of each gender is 1/2.
What is the probability that the other kid is also a girl?

A lot of respondents are adding information that is not in the question.
Many talk about biological chances of a second child being this or that.
This is immaterial. The question gives the parameters.
Each birth has a 1/2 (or 50/50) chance of being a girl or boy.
Second. It does not state (or even imply) that the girl is first (or second) born.
It merely states 2 kids and one is a girl.
Within the framework of the question, the girl is not part of the question (she is a red herring).
The question is; what is the chance of the other (not necessarily the second) kid being a girl.
As we are told that (for this question) the chance of any birth being 50/50, then the answer is a 50% chance of a girl.
The girl/boy, boy/girl argument is mute, because we are not told order of birth, nor are we asked to infer order of birth.
So the girl/boy, boy/girl options are, in fact, only one option.
Therefore the options are two girls or one of each.
And this is purely a 50/50 proposition.

I hope this explanation helps sort out the confusion.

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On 5/27/2018 at 2:38 AM, CSIQ said:

... the girl/boy, boy/girl options are, in fact, only one option.

They are equally likely but they are not the same.
But it's worse than that. The OP is deficient, because it does not tell us how we came to know what we know.

Spoiler

Suppose the person that is telling us about Teanchi and Beanchi has first decided this: If a family has at least one son, I will say the family "has at least one boy." If not, then I will say the family "has at least one girl." But let's say we don't know that is his algorithm for describing families to us. Which is true, because the OP does not disclose it. But what would happen in that case?

We know all two-child families belong, in equal numbers, to these four gender groups: { GG GB BG BB }. If T&B are GB BG or BB, our informant will say T&B have at least one boy. Only in the GG case, where there is not "at least one son," will he say T&B have at least one girl. The probability that the other kid is a girl is then 100%!

Or suppose our informant has first decided this: If the elder of two children is a son, I will say the family "has at least one boy." If not, then I will say the family "has at least one girl." What happens in that case? If T&B are BG or BB, he will say T&B have at least one boy. Otherwise, when T&B are GG or GB, he will say T&B have at least one girl. The probability that the other kid is a girl is then 50%.

Or suppose our informant has decided this: If there is at least one daughter, I will say "the family has at least one girl." Otherwise, I will say "the family has at least one boy." Now, if T&B are GG GB BG he will say T&B have at least one girl, and the probability that the other kid is a girl is 1/3.

Ouch. That is the case that seems most likely, but we can't assume our informant's algorithm with certainty. The OP as stated is deficient.

Instead, let's create a situation where we know how we know what we know, and therefore will let us find the probability that "the other kid is a girl," unambiguously.

Spoiler

If births are gender-neutral, we know that families with exactly two children divide equally into four categories for the sex of their older/younger child. Namely: { GG GB BG BB }. The idea now is to eliminate the last category, but do it in a way that does not inject any bias. If we can do that, then we can say GG is one of three equally likely cases { GG GB BG } so the answer is 1/3.

Here's one way.

Pick a number from a hat containing all number from 1-100, and say it turns out to be 13. Now assemble thousands and thousands of two-children families from the general population. This means equal numbers of GG GB BG and BB families. Now, in some random order, we ask them this question: "Is at least one of your children a girl?" and let's say that the 13th family to answer "Yes" is the Smith family.

We can be certain that the Smith family is type GG GB or BG with equal probability.

Doing that, we know the answer is 1/3.

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