1. **Problem Statement:** The probability that a patient fails to recover from a heart transplant operation is 0.5% (or 0.005). Eight patients are selected at random for the operation. We need to define the random variable, state its distribution with parameters, and write an expression for the probability that less than 3 patients will recover. 2. **Define the random variable:** Let $X$ be the number of patients who fail to recover from the operation out of the 8 selected. 3. **Distribution of the random variable:** Since each patient either fails or recovers independently, and the probability of failure is constant, $X$ follows a binomial distribution. The binomial distribution formula is: $$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where: - $n = 8$ (number of trials/patients), - $p = 0.005$ (probability of failure), - $k$ is the number of failures. 4. **Expression for the probability that less than 3 patients will recover:** The problem asks for the probability that less than 3 patients will recover. Since $X$ counts failures, the number of patients who recover is $8 - X$. Less than 3 patients recover means: $$8 - X < 3 \implies X > 5$$ So, the probability that less than 3 patients recover is: $$P(X > 5) = P(X = 6) + P(X = 7) + P(X = 8)$$ Using the binomial formula: $$P(X > 5) = \sum_{k=6}^{8} \binom{8}{k} (0.005)^k (0.995)^{8-k}$$ This is the required expression. **Final answers:** - Random variable: $X$ = number of patients who fail to recover. - Distribution: $X \sim \text{Binomial}(n=8, p=0.005)$. - Probability expression: $$P(X > 5) = \sum_{k=6}^{8} \binom{8}{k} (0.005)^k (0.995)^{8-k}$$