Trouble working out the Central Limit Theorem

Question

Let $S_{100}$ be the number of heads that turn up in 100 tosses of a fair coin. Use the Central Limit Theorem to estimate $P(S_{100} ≤ 45)$. How do I start off this question?

Answer 1

Hint: You can write $S_{100}$ as the sum of $n=100$ i.i.d Bernoulli$(p=1/2)$ random variables $X_i$ i.e. $$S_{100}=\sum_{i=1}^{100}X_i$$ where $$X_i=\begin{cases}1, & \text{if Heads, } &p=\frac{1}{2}\\ 0, & \text{if Tails, }& q=1-p=\frac{1}{2}\end{cases}$$ Now, find $μ=E[X_i]$ and $σ^2=var(X_i)$ and use the CLT to obtain that $S_{100}$ is approximatelly normally distributed with parameters $μ_{100}=100μ$ and $σ_{100}^2=100σ^2$, in symbols $$S_{100}\sim N(100\cdot μ, 100 \cdot σ^2)$$ approximately.