Chapter 3, Problem 3.30TE

In Laplaces rule of succession (Example 5e), suppose that the first n flips resulted in r heads and $n-r$ tails. Show that the probability that the $\left(n+1\right)$ flip turns up heads is $\frac{\left(r+1\right)}{\left(n+2\right)}$. To do so, you will have to prove and use the identity ${\displaystyle {\int }_0^1{y}^n{\left(1-y\right)}^{m}dy=\frac{n!m!}{\left(n+m+1\right)!}}$

Hint: To prove the identity, let $C\left(n,m\right)={\displaystyle {\int }_0^1{y}^n{\left(1-y\right)}^{m}dy}$. Integrating by parts yields $C\left(n,m\right)=\frac{m}{n+1}C\left(n+1,m-1\right)$.

Starting with $C\left(n,0\right)=1/\left(n+1\right)$, prove the identity by induction on m.