I have a Bernoulli trial with success rate $p$ and failure rate $1-p$ the odds of $k$ successes is $\binom{N}{k} p^k (1-p)^{N-k}$. I need to evaluate an integral
$$ \int\_0^1 dp p^k (1-p)^{N-k} = \frac{k!(N-k)!}{(N+1)!} $$
This was done with Mathematica, but you can use induction. To avoid mindlessly integrating, **can you read this integral as an expectation and evaluate it probabilistically**? Maybe after rescaling.


Maybe there's a name for Bernoulli trials where the success rate $p$ uniformly random.



How to reading of an integral?  Bernoulli trials with variable success rate, p
[More or less, yes](https://math.stackexchange.com/questions/3528/beta-function-derivation).


