1. **Problem Statement:** We have a binomial distribution with parameters $n=20$ and $p=0.4$. We want to decide whether to use the binomial distribution directly or approximate it with a normal distribution to calculate probabilities. 2. **Relevant Formulas and Rules:** - The binomial distribution is given by $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where $X$ is the number of successes. - The normal approximation to the binomial is used when $n$ is large enough such that both $np$ and $n(1-p)$ are greater than or equal to 10. This ensures the binomial distribution is approximately symmetric and bell-shaped. 3. **Check Conditions for Normal Approximation:** Calculate: $$np = 20 \times 0.4 = 8$$ $$n(1-p) = 20 \times 0.6 = 12$$ 4. **Interpretation:** - Since $np = 8$ is less than 10, the normal approximation is not recommended. - The binomial distribution should be used directly for accurate probability calculations. **Final conclusion:** Use the binomial distribution directly because $np$ is less than 10, so the normal approximation is not appropriate here.