1. **Problem statement:** A gardener planted 20 tomato seeds with a 50% chance each seed will germinate. 2. **Relevant formula:** This is a binomial probability problem where the number of trials $n=20$, probability of success $p=0.5$, and number of successes $k$ varies. The binomial probability formula is: $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. 3. **Part (a): Probability exactly 10 seeds germinate** Calculate $P(X=10)$: $$P(X=10) = \binom{20}{10} (0.5)^{10} (0.5)^{10} = \binom{20}{10} (0.5)^{20}$$ Calculate $\binom{20}{10}$: $$\binom{20}{10} = \frac{20!}{10!10!} = 184,756$$ So, $$P(X=10) = 184,756 \times (0.5)^{20} = 184,756 \times \frac{1}{1,048,576} \approx 0.176$$ 4. **Part (b): Probability at least 15 but not more than 18 seeds germinate** Calculate $P(15 \leq X \leq 18) = P(X=15) + P(X=16) + P(X=17) + P(X=18)$ Each term: $$P(X=k) = \binom{20}{k} (0.5)^{20}$$ Calculate each binomial coefficient: - $\binom{20}{15} = 15,504$ - $\binom{20}{16} = 4,845$ - $\binom{20}{17} = 1,140$ - $\binom{20}{18} = 190$ Sum: $$\sum = 15,504 + 4,845 + 1,140 + 190 = 21,679$$ Probability: $$P = 21,679 \times (0.5)^{20} = 21,679 \times \frac{1}{1,048,576} \approx 0.0207$$ 5. **Part (c): Mean number of seeds that germinate** The mean (expected value) for a binomial distribution is: $$\mu = np = 20 \times 0.5 = 10$$ **Final answers:** - (a) $P(X=10) \approx 0.176$ - (b) $P(15 \leq X \leq 18) \approx 0.0207$ - (c) Mean number of seeds germinating $= 10$