How likely is it that this child is a girl?

Question

Here's a fun puzzle. You're going to go have dinner at your teacher's house. You know he has three kids, but you don't know anything else about them.

Once you get to his house, a boy answers the door. He says "my dad will be downstairs in just a second; you can wait in the kitchen." You wait as you're told, and soon your teacher comes downstairs. He hands you a beverage and says "my oldest child, my daughter Jane, should be home soon."

So, you've met one of his children, who happened to be a boy. You also know that the oldest of his three children is a girl. This means that he has one other child about which you know nothing. What is the probability that this other child is a girl?

(It's not 50%, if that's what you're wondering.)

Answer 1

Short answer: 66.67%.

Long answer:
Take the information the problem gives you and narrow it down. First thing you find out is that he's got three kids. There are eight possible ways to have three kids: GGG, GGB, GBG, GBB, BBB, BGG, BGB, BBG. Each of these is equally likely.

Next you meet a boy, so all you know is that he's got a three-child family with at least one boy. This eliminates the possibility of GGG. The remaining seven options are all equally likely.

Then you find out that his first-born was a girl, so this eliminates four of the seven options. The remaining three possibilities are GBG, GBB, and GGB. Each is equally likely.

In two of the three possible scenarios, your teacher has two daughters and one boy. In one of the three, your teacher has two boys and one girl. Therefore, there's a 66.67% chance that the child you know nothing about is a girl.

Answer 2

Actually, it IS 50%. From the way you describe the problem (i.e. "one of his children happened to be a boy"), it sounds like you intended to convey a variant of that hoary old chestnut which goes something like this:

"John has two children. His oldest is a boy. What is the probability that the other one is also a boy? (Answer: 1/2)

John has two children. One of them is a boy. What is the probability that the other one is also a boy? (Answer: 1/3)"

The difference between the two situations is that in one case, you are given information about a specific child whereas in the second you are given information about the pair of children as a whole. This is a relevant distinction: in the first case, if the guy has exactly one male child and exactly one female child, there is only a 50% chance that the condition in the problem - namely, that "the oldest child is a boy" - is true. Conversely, the same case gives a 100% chance of "one of them is a boy" is true.

Now, from the way you lay out the problem, you probably think that you have said "a specific child is a girl, and one of the other children is a boy" - from which we could deduce the probability of the remaining child being a girl is 2/3 (the specific child's gender has no effect on the others, and of the three assignments consistent with one of the remaining children being a boy (BB, BG, GB), two of them have the other remaining child be a girl).

Here's the problem: you did not say "one of the other children is a boy." The problem specified that the specific child who happened to answer the door is a boy. If there are two younger children, and one is a boy and the other is a girl, there is a 50% chance that it would have been the girl who answered the door and the conditions in the problem would no longer hold. Thus, of the possibilities for the gender of the two younger children (BB, GG, BG, GB), and we assume without loss of generality that the first one answers the door, we have only BB and BG consistent - i.e. GB is not consistent, and the probability is 1/2, not 2/3.

Answer 3

About 51%. More boys are born, but girls are stronger, so there is a larger number of girls in the world.

Unless you're referring to the father having said, "my daughter Jane" instead of simply saying "my daughter", which may tend to suggest that he has more than one daughter. In that case it would be 100%, obviously. But stating the daughters name isn't necessarily indicative that she is not the only one.

Answer 4

It should be 50%, unless the Dad has some bias in his sperm. The sex of each child is completely unrelated to it's siblings. Instead, it is related to the ratio of x/y sperm from the father. Assuming he is a typical XY male and meiosis proceeds normally, it should be a 50% chance.

Answer 5

It has to be 50%

Answer 6

No genuine thank you to tell. could purely be success. My grandmother produced 8 females, miscarried a series of twins (a million boy and a million female) My mom had 3 females and a million boy My sister has 2 females i'm pregnant with a woman. perhaps next time i gets a boy lol.

Answer 7

0% because if he had another daughter he would have said "my oldest daughter should be home soon." However, he said "my oldest child, my daughter..." which leads me to believe that he only has 1 daughter and 2 sons.

Answer 8

it's about 25 %

Answer 9

its below 50%

Answer 10

50% is the proper answer.

But thanks to some bias in his sentence "my oldest child, MY DAUGHTER Jane", you can *guess* that his other child is a son.