Question

How to calculate probability that such a family consists of at least one male child and at least one female child

Answer 1

Let's break this down into three possibilities:

Only male children

Only female children

A mixture of male and female children

The three groups sum to 1 (because all families have at least one child), and so if we compute the probability of the first two groups, we can subtract that from one and get our answer. This is even easier because the two are symmetric--the probability of only having male children is the same as the probability of only having female children.

So how would we get the probability of only having sons?

Start with the probability of having a family with exactly one son (and zero daughters). The chance that the family size (FF) is 1 is:

P(F=k)=0.5k→P(F=1)=0.5P(F=k)=0.5k→P(F=1)=0.5

The probability that a family of size 1 has only sons (S=FS=F) is:

P(S=k|F=k)=0.5k→P(S=1|F=1)=0.5P(S=k|F=k)=0.5k→P(S=1|F=1)=0.5

But we want an unconditional probability, so

P(S=1,F=1)=P(S=1|F=1)P(F=1)=0.25P(S=1,F=1)=P(S=1|F=1)P(F=1)=0.25.


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