# What's the probability that Erica has one boy and one girl?

During a flight on an airplane, Eric strikes up a chat with Erica, the person sitting next to him. It turns out that Erica has two kids, and at least one of them is a girl born on a Tuesday. Being a mathematician, Eric decides to find the probability that Erica has a boy and a girl before asking her. What is the probability that Erica has one boy, and one girl? Assume an equal chance of giving birth to either gender and an equal chance to giving birth on any day.

I was looking at the solution to this question and for some reason they were looking at the number of pairs of $bg$ in a two week time period. Why not a week period?

• See the diagram posted. – user19405892 Dec 1 '15 at 4:25
• It is a one week period, just that each column is defined by both the gender and the day of the week. – Element118 Dec 1 '15 at 4:25
• They are not considering a two-week period, rather they are considering the 14 possibilities for each child (7 possibilities for day of the week born, 2 possibilities for gender). – kccu Dec 1 '15 at 4:26

The possibilities are:

• $7 \times$ first child is a boy born any day and second child is a girl born on Tuesday.
• $7 \times$ first child is a girl born Tuesday and second child is a boy born on anyday.
• $6 \times$ first child is a girl born any other day and second child is a girl born on Tuesday.
• $6 \times$ first child is a girl born Tuesday and second child is a girl born on any other day.
• $1 \times$ both children are girl born Tuesday.

These are marked out as the yellow cross on the grid; which is first child's gender and birth-weekday versus second child's gender and birth-weekday.

tl;dr: Notice the second column and row grouping weekday with gender.

Of course, there's a data collection issue here.   The above assumes that the information provided was of the form: "at least one of my two children was a girl born on a Tuesday."   However, this is unlikely to be the quality of information conveyed during a casual introduction.