1. **Problem Statement:** We want to approximate a binomial distribution $S_n \sim \text{Bin}(n, p)$ using a normal distribution when $n$ is large. 2. **Theorem Used:** The DeMoivre-Laplace Limit Theorem states that for $p \in [0,1]$ and $S_n \sim \text{Bin}(n, p)$, $$\lim_{n \to \infty} \Pr\left\{ a \leq \frac{S_n - np}{\sqrt{np(1-p)}} \leq b \right\} = \Phi(b) - \Phi(a),$$ where $\Phi$ is the cumulative distribution function (CDF) of the standard normal distribution. 3. **Key Idea:** The binomial distribution can be approximated by a normal distribution with mean $\mu = np$ and variance $\sigma^2 = np(1-p)$ when $np(1-p) \geq 10$. 4. **Approximation Formula:** $$S_n \approx N\left(np, np(1-p)\right).$$ 5. **Important Rules:** - Use the continuity correction when approximating discrete binomial probabilities by continuous normal probabilities. For example, $$\Pr(S_n \leq k) \approx \Pr\left(Z \leq \frac{k + 0.5 - np}{\sqrt{np(1-p)}}\right),$$ where $Z \sim N(0,1)$. - Check that $np(1-p) \geq 10$ to ensure the normal approximation is valid. 6. **Summary:** To approximate binomial probabilities: - Calculate mean $\mu = np$ and standard deviation $\sigma = \sqrt{np(1-p)}$. - Convert binomial variable $S_n$ to standard normal variable $Z = \frac{S_n - \mu}{\sigma}$. - Use standard normal tables or software to find probabilities. This method simplifies calculations for large $n$ and moderate $p$ values.