Seventy-six percent of adults want to live to age 100 . You randomly select five adults and ask them whether they want to live to age 100 . The random variable represents the number of adults who want to live to age 100. Complete parts (a) through (c) below. (a) Construct a binomial distribution. \begin{tabular}{|c|c|} \hline$x$ & $P(x)$ \\ \hline 0 & $\square$ \\ \hline 1 & $\square$ \\ \hline 2 & $\square$ \\ \hline 3 & $\square$ \\ \hline 4 & $\square$ \\ \hline 5 & $\square$ \\ \hline \end{tabular} (Round to five decimal places as needed.)

Step 1 ：The problem is asking for a binomial distribution. A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials with the same probability of success. In this case, the Bernoulli trial is whether an adult wants to live to age 100, and the probability of success is 76% or 0.76. The fixed number of trials is 5, as we are selecting five adults.

Step 2 ：The probability mass function of a binomial distribution is given by the formula: \[P(x) = C(n, x) * p^x * (1-p)^(n-x)\] where: - C(n, x) is the number of combinations of n items taken x at a time, - p is the probability of success, - n is the number of trials, and - x is the number of successes.

Step 3 ：We can use this formula to calculate the probabilities for x = 0, 1, 2, 3, 4, 5. Here, n = 5 and p = 0.76.

Step 4 ：The calculated probabilities are: 0.0008, 0.01261, 0.07985, 0.25285, 0.40035, 0.25355.

Step 5 ：The final binomial distribution is as follows: \begin{tabular}{|c|c|} \hline$x$ & $P(x)$ \ \hline 0 & \boxed{0.0008} \ \hline 1 & \boxed{0.01261} \ \hline 2 & \boxed{0.07985} \ \hline 3 & \boxed{0.25285} \ \hline 4 & \boxed{0.40035} \ \hline 5 & \boxed{0.25355} \ \hline \end{tabular}