In Laplace’s rule of succession (Example 5e), suppose that the first nflips resulted in rh...

In Laplace’s rule of succession (Example 5e), suppose that the first nflips resulted in rheads and n − rtails. Show that the probability that the (n+ 1) flip turns up heads is (r + 1)/(n + 2). To do so, you will have to prove and use the identity

Integrating by parts yields

Starting with C(n, 0) − l/(n + 1), prove the identity by induction on m.