To construct a binomial distribution for the given problem, we need to determine the probability of each possible outcome for [tex]\( x \)[/tex], where [tex]\( x \)[/tex] is the number of adults out of five who want to live to age 100.
Given data:
- [tex]\( n = 5 \)[/tex] (number of trials, i.e., number of adults selected)
- [tex]\( p = 0.72 \)[/tex] (probability of success, i.e., the probability that an adult wants to live to age 100)
The binomial probability mass function (pmf) is given by:
[tex]\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \][/tex]
Here, we are given the probabilities for each possible value of [tex]\( x \)[/tex], where [tex]\( x \)[/tex] ranges from 0 to 5. Let's fill in the table with these values.
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $P(x)$ \\
\hline
0 & 0.00172 \\
\hline
1 & 0.02213 \\
\hline
2 & 0.11380 \\
\hline
3 & 0.29263 \\
\hline
4 & 0.37623 \\
\hline
5 & 0.19349 \\
\hline
\end{tabular}
\][/tex]
Each value of [tex]\( P(x) \)[/tex] needs to be rounded to five decimal places, and the values given already satisfy this requirement.
Hence, the completed binomial distribution table is:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $P(x)$ \\
\hline
0 & 0.00172 \\
\hline
1 & 0.02213 \\
\hline
2 & 0.11380 \\
\hline
3 & 0.29263 \\
\hline
4 & 0.37623 \\
\hline
5 & 0.19349 \\
\hline
\end{tabular}
\][/tex]
