# Which family do you think is likely to have a girl ?

You along with your friend are standing in front of two houses. Each of those houses inhabits a family with two children.

Your friend tells you the below two facts:

1) On your left is a family that has a boy who likes accounts but the other child loves science.

2) On the right is a family with a seven year old boy and a new born baby.

You ask him, “Does either of the family have a girl?”

To this he replies, “I am not quite sure. But can you guess that? If you are right, I will give you $200.”

Which family do you think is likely to have a girl ?

This is a poorly stated problem. The existing answer makes assumptions that aren’t stated, which result in the conclusion that the information about which child is older is relevant. It’s not.

Let’s look at the problem another way. In the house on the left, we have:

Accounts / Science

Boy Boy

Boy Girl

In the house on the right we have:

Younger / Older

Girl Boy

Boy Boy

With the given information, the odds are the same for either house to have a girl.

Consider a similar scenario with coins. I flip two pennies and two dimes. I tell you that the first dime flipped was heads and that the shinier penny is also heads. In each case I’ve distinguished the two events–it doesn’t matter what I use to distinguish them, it only matters that they have been distinguished.

In order for the information about which dime was heads to be useful, you have to make an assumption regarding the complete information available to me and *why* I selected to share order information about the dime and not the penny, as well as *why* selected to share information about shininess about the penny. If I know the result for both dimes, then the information about which was flipped first is useless: I would not revel information about a coin being tails, so I would always select to tell about the dime that came up heads. Same logic applies to the pennies. Since the selection bias is to only tell about heads, the information about order or shininess is irrelevant because it is revealed AFTER the selection of which dime to reveal is made. Information about order, shininess, etc. is only useful if it is ARBITRARILY revealed.

There is a more relevant piece of information revealed, which is that your friend is “not quite sure” if either family has a girl. In order for this to be true, two things must be true: your friend does not know the gender of all four children and none of the children your friend does know about is a girl. This leaves open the possibility that your friend knows the gender of one of the two other children (either the child that likes science or the baby), and that child is a boy. Given that your friend knows that the other child in the house on the left likes science, I’m going to assume it is more likely that he also knows the gender of the second child in the house on the left, than the gender of the baby. Based on THIS information, I pick the house on the right.

1/2

In the house on the left, there are three possibilities:

Younger Older

Girl Boy

Boy Girl

Boy Boy

We cant have a girl-girl option because it has been mentioned that there is at least one boy in that house. Now all these outcomes are equally likely, and we have two events with girls, the chances of having a girl in the left house is 2/3.

In the house on the right, there are two possibilities because we already know that the older child is a boy:

Younger Older

Girl Boy

Boy Boy

Here the chances are 1/2.

Thus you must choose the house on the left for better chances at winning.

### Your Answer

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