This is a binomial distribution problem where:

• $n = 5$ (the number of trials, since we are selecting 5 adults)
• $p = 0.68$ (the probability of success, i.e., an adult wants to live to age 100)
• $q = 1 - p = 0.32$ (the probability of failure, i.e., an adult does not want to live to age 100)

The probability mass function for a binomial distribution is given by:

$P(x) = \binom{n}{x} \cdot p^x \cdot q^{n-x}$

where $\binom{n}{x} = \frac{n!}{x!(n-x)!}$ is the binomial coefficient.

We will calculate $P(x)$ for $x = 0, 1, 2, 3, 4, 5$.

Let me compute the probabilities for each value of $x$.Here is the binomial distribution for $n = 5$ and $p = 0.68$:

\hline x & P(x) \\ \hline 0 & 0.00336 \\ 1 & 0.03565 \\ 2 & 0.15152 \\ 3 & 0.32198 \\ 4 & 0.34210 \\ 5 & 0.14539 \\ \hline \end{array}$$ These values are rounded to five decimal places. Would you like further details on the calculations or have any questions? Here are 5 related questions: 1. How do we calculate the expected value of a binomial distribution? 2. What is the variance of this binomial distribution? 3. Can we approximate this binomial distribution with a normal distribution? 4. What happens if we change the probability $$p$$ to 0.5? 5. How would the distribution change if we surveyed 10 people instead of 5? **Tip:** In binomial distributions, as the number of trials increases, the distribution tends to become more symmetric, especially when $$p$$ is closer to 0.5.