I have two children and a son who was born on a Thursday. What is the probability that I have a daughter? Let's denote the days of the week with the numbers \(1, 2, ... , 7\) with \(1\ =\) Monday, \(2 =\) Tuesday and so on. Now we can define the event as "a boy was born on day \(n\) " as \(B_n\), and similarly for \(G_n\).

For example, \(B_3\) means a boy born on a Wednesday and \(G_1\) means a girl born on a Monday. With this notation, we can write events like: \(B_3G_1\) means: "The first child was a boy born on a Wednesday and the second child was a girl born on a Monday".

Let's assume that the probability of a child being born on each day of the week is the same (which, like the sex of the child, is not entirely true, but is a reasonable assumption to keep the problem simple). This leads to the following \(27\) equally likely ways my children could have been born(\(B_4\) represents a boy born on Thursday):

$$

\begin{matrix}

B_4G_1 & G_1B_4 & B_1B_4 & B_4B_1\\

B_4G_2 & G_2B_4 & B_2B_4 & B_4B_2\\

B_4G_3 & G_3B_4 & B_3B_4 & B_4B_3\\

B_4G_4 & G_4B_4 & B_4B_4 & \\

B_4G_5 & G_5B_4 & B_5B_4 & B_4B_5\\

B_4G_6 & G_4B_4 & B_6B_4 & B_4B_6\\

B_4G_7 & G_7B_4 & B_7B_4 & B_4B_7

\end{matrix}

$$

The table above is a complete list of all the ways in which two children can be born if at least one of them is a boy born on a Thursday. Under our assumptions, of course, all these events are equally likely. There are \(27\) possibilities from which \(14\) (those in the first two columns) contain a girl, so the probability that I have a daughter is \(14/27\approx 51.9\% \neq 50\%\).