Sum of a part of binomial expansion.

Question

The calculation of $\sum_{k=0}^n \binom{n}{k}p^k (1-p)^{n-k}$ can be simply done with $(p+1-p)^n$. But is there any closed form formula for $\sum_{k=i}^n \binom{n}{k}p^k (1-p)^{n-k}$? If that is not solvable, is there a closed form formula when $i=\frac{n}{2}$?

Answer 1

You're looking for the regularised incomplete beta function.