The King's Party - Detailed Calculation

Let T be the event that a man has a girl child born on a Tuesday. Let G be the event that at least on of the children is a girl. Let GB be the event that one child is a girl and the other is a boy.

For all men with exactly 2 children, we know that:

P(G) = ¾               (That is, the probability that at least one child is girl is ¾ )

P(GB/G) = 2/3    (As we have already seen. That is, the probability of the other child being a boy, given that one is a girl, is 2/3)

P(GB) = ½ (That is, the probability that one child is a girl and the other is a boy, is ½)

P(GG) = 1/4

Now, if one child is a girl and the other is a boy, what is the probability that the girl was born on a Tuesday? Well, the girl could have born on any day with equal probability. It is only one in a seven chance that she may be born on a Tuesday. Hence,

P(T/GB) = 1/7

And if both the children are girls? What is the probability that the man has at least one girl born on a Tuesday?

P(T/GG) = (1/7) + (1/7) – (1/7)*(1/7) = 2/7 – 1/49 = (14 – 1)/49 = 13/49

So, overall, what is the probability that a man has a girl born on a Tuesday? Well, we add up the two cases.

P(T) = P(GB).P(T/GB) + P(GG).P(T/GG) = (½)(1/7) + (1/4)(13/49) = (14+13)/(49*4) = 27/196

Now, the real question. 

What is the probability that the man has one boy and one girl (event GB) given that the man has a girl child born on a Tuesday (event T)? So, we have to find out the probability of event GB, given the event T.

Using Bayes’ theorem:

P(GB/T) = P(T/GB).P(GB)/P(T)

P(GB/T) =  =  = 14/27 ≈ 0.52

It so turns out that if we know that the girl was born on a Tuesday, the chance that the other child is a boy is 0.52, which is very close to ½ . Intuitively, it will get closer to ½, if we know the date of birth of the girl and get even closer if we also know her name! If you know the girl, or meet her, the probability that she has a brother goes to exactly ½. (And this last statement actually makes sense intuitively).