# The man with two boys [duplicate]

I have recently seen a probability question which says
"i am asking randomly the persons I met if they are having two chidren and one of them is a boy who was born on tuesday. At last I met one whose answer is yes. What is the probability that the other child is also a boy. Assume equal probability to either gender and equal probability to be born on each day of the week"

I could actually solve it to 2/21. Did I do it right or can some one help me solve it?

## marked as duplicate by Alex Becker, Cheerful Parsnip, azimut, Cameron Buie, Zev ChonolesMay 21 '13 at 7:53

• – Zev Chonoles May 21 '13 at 7:26

This is going to get a bit long for a comment.

Suppose you are asking three questions in sequence:

1 Do you have exactly two children?

2 Do you have a child who is a boy born on a Tuesday?

If the answer is yes, the person continues in the sample - call this $BT$ and you ask

3 Are both of your children boys?

Say the yes answers to this are $BB$, you are looking for $\frac {BB}{BT}$

Note that (given reasonable assumptions) one fourteenth of all children are tuesday-born boys. However, at stage 2, there will be $0$ or $1$ or $2$ Tuesday-born boys in the family. Case $0$ is discarded, and the other two are not equally likely.

Taking the day and the sex into account, there are 14 possibilities for each child - 196 altogether. In one of these, both are Tuesday-boys. There are 26 more possibilities containing a Tuesday-boy, making 27 in all (the $2$ case takes two of the expected 28) - twelve of which have two boys (two of the fourteen having been taken into account already), and fourteen (as expected) with a boy and a girl.

Counting, one sees the proportion with two boys is $\frac {13}{27}$.

I noted this, because it was not explained in the answers to the closely related question Zev Chonoles linked.