68 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.
Question content area bottom
Part 1
​(a) Construct a binomial distribution.
x
​P(x)
0

enter your response here
1

enter your response here
2

enter your response here
3

enter your response here
4

enter your response here
5

enter your response here
​(Round to five decimal places as​ needed.)

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 \( p = 0.68 \):

\[
\begin{array}{|c|c|}
\hline
x & P(x) \\
\hline
0 & 0.00336 \\
1 & 0.03565 \\
2 & 0.15152 \\
3 & 0.32198 \\
4 & 0.34210 \\
5 & 0.14539 \\
\hline
\end{array}
\]

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.

68 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.

Learn how to construct a binomial distribution based on a survey where 68% of adults want to live to age 100. This example walks through the probability calculation for different outcomes using the binomial theorem. Suitable for college-level students studying probability and statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation