1. **Problem statement:** Given a binomial random variable $X$ with expected value $E(X) = 6$ and variance $Var(X) = 2.4$, find the probability $P(X=5)$. 2. **Recall the formulas for a binomial distribution:** - Expected value: $$E(X) = np$$ - Variance: $$Var(X) = np(1-p)$$ where $n$ is the number of trials and $p$ is the probability of success in each trial. 3. **Use the given values to find $n$ and $p$:** From $E(X) = np = 6$ and $Var(X) = np(1-p) = 2.4$. 4. **Express $p$ in terms of $n$:** $$p = \frac{6}{n}$$ 5. **Substitute $p$ into the variance equation:** $$2.4 = n \times \frac{6}{n} \times \left(1 - \frac{6}{n}\right) = 6 \left(1 - \frac{6}{n}\right)$$ 6. **Simplify and solve for $n$:** $$2.4 = 6 - \frac{36}{n}$$ $$\frac{36}{n} = 6 - 2.4 = 3.6$$ $$n = \frac{36}{3.6} = 10$$ 7. **Find $p$:** $$p = \frac{6}{10} = 0.6$$ 8. **Calculate $P(X=5)$ using the binomial probability formula:** $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ 9. **Substitute values:** $$P(X=5) = \binom{10}{5} (0.6)^5 (0.4)^5$$ 10. **Calculate the binomial coefficient:** $$\binom{10}{5} = \frac{10!}{5!5!} = 252$$ 11. **Calculate powers:** $$(0.6)^5 = 0.07776$$ $$(0.4)^5 = 0.01024$$ 12. **Calculate the probability:** $$P(X=5) = 252 \times 0.07776 \times 0.01024 \approx 0.2007$$ **Final answer:** $$\boxed{P(X=5) \approx 0.2007}$$