Multiple i.i.d. draws from Normal distribution

Question

Say you have $N$ i.i.d. draws of a normal distribution with parameters $\mu$ and $\sigma$. What is the probability that $k$ of those draws are larger than some value $Y$? Is my answer correct?

Answer 1

For the exact $k$ draws your answer is correct. Namely, let $X_1,...,X_N$ be i.i.d from $N(\mu, \sigma^2)$, and let $W$ be the number of draws that are larger then $y$, thus $W\sim \operatorname{Bin}(N, p)$, where
$$ p = P(X > y ) = 1 - \Phi\left( \frac{y - \mu}{\sigma} \right), $$ hence for exactly $k$ you have $$ P(W=k) = \binom{N}{k}p^k ( 1 - p)^{n - k}, $$ and for "at least $k$" $$ P(W \ge k) = \sum_{i=k}^N \binom{N}{i}p^i ( 1 - p)^{n - i}. $$