Forgive me for my notation and lack of formatting prowess, it's been a while since I've done any of this.
We have flipped a biased coin (unknown weight) $n$ times and received $k$ successes (or heads). I am trying to calculate the probability that there be a success on flip $N+1$
If we assume the bias $p$ is sampled uniformly (and is constant throughout the flip), then is the correct way to calculate this:
$\int P(p| X)*p \partial p$ where $X$ is n successes and k failures
thus
$P(p| X) = \frac{P(X|p)*f(p)}{\int_{0}^{1} P(X|p) \partial p}$
Assuming a uniform prior for $p$ and that ${\int_{0}^{1} P(X|p)} = \frac{1}{n+1}$, we get
$\int_{0}^{1} P(X|p)*(n+1)*p \partial p $
where $P(X|p)$ is the binomial probability.
Is this correct?