To construct the binomial distribution, we need to use the following formula for the probability of exactly $x$ successes in $n$ trials:

$P(x) = \binom{n}{x} p^x (1 - p)^{n - x}$

Where:

• $n$ = 5 (the number of trials, which is the number of adults selected),
• $p$ = 0.80 (the probability that an adult wants to live to age 100),
• $1 - p$ = 0.20 (the probability that an adult does not want to live to age 100),
• $x$ is the number of adults who want to live to age 100.

For each value of $x$, we'll calculate $P(x)$ using the formula.

Step 1: Calculate each $P(x)$ for $x = 0, 1, 2, 3, 4, 5$.

For $x = 0$:

$P(0) = \binom{5}{0} (0.80)^0 (0.20)^5 = 1 \times 1 \times 0.00032 = 0.00032$

For $x = 1$:

$P(1) = \binom{5}{1} (0.80)^1 (0.20)^4 = 5 \times 0.80 \times 0.0016 = 0.0064$

For $x = 2$:

$P(2) = \binom{5}{2} (0.80)^2 (0.20)^3 = 10 \times 0.64 \times 0.008 = 0.0512$

For $x = 3$:

$P(3) = \binom{5}{3} (0.80)^3 (0.20)^2 = 10 \times 0.512 \times 0.04 = 0.2048$

For $x = 4$:

$P(4) = \binom{5}{4} (0.80)^4 (0.20)^1 = 5 \times 0.4096 \times 0.20 = 0.4096$

For $x = 5$:

$P(5) = \binom{5}{5} (0.80)^5 (0.20)^0 = 1 \times 0.32768 \times 1 = 0.32768$

Step 2: Fill out the table

$x$	$P(x)$
0	0.00032
1	0.00640
2	0.05120
3	0.20480
4	0.40960
5	0.32768

Answer:

• $P(0) = 0.00032$
• $P(1) = 0.00640$
• $P(2) = 0.05120$
• $P(3) = 0.20480$
• $P(4) = 0.40960$
• $P(5) = 0.32768$

Would you like any further explanation or details on any of the steps?

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Here are 5 related questions to expand the topic:

1. How do you calculate the binomial coefficient $\binom{n}{x}$?
2. How would the probabilities change if the percentage of adults who want to live to age 100 were 90% instead of 80%?
3. How do you find the mean and standard deviation for a binomial distribution?
4. What happens to the binomial distribution if the number of adults surveyed (n) increases to 10?
5. How can you use a cumulative binomial distribution table for this problem?

Tip: For more complex binomial distributions, consider using a calculator or software to quickly compute the probabilities!