https://file.aiquickdraw.com/mathbot/file/1730911640306qf17gyyh.jpg

To solve this problem, let's go over each part of the question:

### Given Data
- Probability of a person donating blood (\( p \)) = 0.21
- Number of trials (\( n \)) = 4

The problem involves finding the mean and variance of a binomial distribution, which we can calculate using the following formulas:

### Part (a): Mean of the Binomial Distribution
The mean (\( \mu \)) of a binomial distribution is calculated using:
\[
\mu = n \times p
\]
Substituting the values:
\[
\mu = 4 \times 0.21 = 0.84
\]
So, the mean \( \mu \) is **0.84**.

### Part (b): Variance of the Binomial Distribution
The variance (\( \sigma^2 \)) of a binomial distribution is calculated using:
\[
\sigma^2 = n \times p \times (1 - p)
\]
Substituting the values:
\[
\sigma^2 = 4 \times 0.21 \times (1 - 0.21)
\]
\[
\sigma^2 = 4 \times 0.21 \times 0.79
\]
\[
\sigma^2 = 0.6636
\]
Rounded to the nearest hundredth, the variance \( \sigma^2 \) is **0.66**.

### Summary of Answers
- **Mean \( \mu \) = 0.84**
- **Variance \( \sigma^2 \) = 0.66**

Would you like further details on the binomial distribution or have any questions?

### Related Questions
1. What is the standard deviation of this binomial distribution?
2. How would the mean and variance change if you sampled 10 people instead of 4?
3. Can you calculate the probability of exactly 2 people donating blood in this scenario?
4. How does changing the probability \( p \) affect the mean and variance?
5. What is the probability that no one in the sample donates blood?

### Tip
For any binomial distribution, remember that the mean and variance are directly influenced by both the number of trials and the probability of success.

Twenty-one percent of people in a city eligible to donate blood actually do. You randomly select four eligible blood donors and ask them if they donate blood. Complete parts (a) through (d) below:

(a) Find the mean of the binomial distribution.
(b) Find the variance of the binomial distribution.

Learn how to calculate the mean and variance of a binomial distribution given a 21% success rate and a sample size of four. This example covers key probability concepts, including formulas for binomial mean and variance, perfect for advanced high school or college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation