Normal approximation to binomial

The idea

When $n$ is large and $p$ is close to $0.5$, $B(n, p)$ is approximated by $N(np, np(1 - p))$. Because the binomial is discrete, use a continuity correction: $P(X \le 45)$ becomes $P(Y < 45.5)$.

1. Check it is suitable: $n$ large, $p$ near $0.5$.
2. Mean $np$, variance $np(1 - p)$.
3. Apply the continuity correction, then find the Normal probability.

Watch out Move the boundary by $0.5$ in the direction that keeps the integers you want.