Probability question Vicotorian england

Question

In Victorian England, the probability of a child born being male was 0.512. In a family of 10 children, find the probability that

• exactly 6 boys
• There were no boys
• There were more girls than boys

Answer 1

If you have never taken and are not taking, a basic probability course, where in the world did you get this question? You need to know that is a given event has probability p of happening then the probability of it happening n times is a row is p multiplied by itself n times: p^n.

The probability of it NOT happening is 1- p and the probability if it happening n times in a row, then not happening m times in a row, in that order, is p^n(1-p)^m.

The probability of it happening n times and not happening m times in any specific order, is also p^n(1-p)^m.

The probability of it happening n times and not happening m times in any order is that probability times the number of different possible orders of m+ n things, m the same, and n the same, is the "binomial coefficient, $\begin{pmatrix}n+ m// n\end{pmatrix}= \begin{pmatrix}m+ m \\ m\end{pmatrix}= \frac{(n+m)!}{n!m!}$. So the probability of it happening n times and not happening m times in any order is $\frac{(n+m)!}{n!m!}p^n(1-p)^m$.