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The question asks for the probability distribution of the number of illiterate people out of a random sample of **eight** people, where about **20%** of people in the population are illiterate. This scenario describes a **binomial distribution** with parameters:

- \( n = 8 \) (the number of trials, or people in the sample),
- \( p = 0.2 \) (the probability that any given person is illiterate).

A **binomial distribution** gives the probability of having exactly \( r \) successes (in this case, illiterate people) in \( n \) independent trials, where each trial has a success probability \( p \).

We can identify the correct histogram by considering the shape of the binomial distribution for these parameters.

1. The expected value \( \mu \) (mean) of the distribution is:
   \[
   \mu = n \times p = 8 \times 0.2 = 1.6
   \]
   This means we expect, on average, about 1.6 illiterate people in the sample.

2. The variance \( \sigma^2 \) is given by:
   \[
   \sigma^2 = n \times p \times (1 - p) = 8 \times 0.2 \times 0.8 = 1.28
   \]
   The standard deviation \( \sigma \) is the square root of the variance:
   \[
   \sigma = \sqrt{1.28} \approx 1.13
   \]
   This indicates the spread of the distribution.

### Key Observations:
- Since the mean is 1.6, the peak of the distribution should be around \( r = 1 \) or \( r = 2 \).
- The distribution will be skewed to the right (since \( p < 0.5 \)), meaning there will be fewer probabilities for larger values of \( r \) (i.e., more than 4 illiterate people).

### Histogram Analysis:
The histogram that fits this description should:
- Peak around \( r = 1 \) or \( r = 2 \),
- Have a long right tail (lower probabilities for \( r \) greater than 4).

Based on this, the **correct histogram** is the **bottom left histogram**, where the peak is at \( r = 1 \) and \( r = 2 \), with the distribution trailing off towards higher values.

Would you like further details on the reasoning? 

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Here are 5 related questions to explore:

1. What is the formula for calculating binomial probability \( P(r) \) for a given \( r \)?
2. How does the shape of the binomial distribution change as the value of \( p \) increases?
3. What is the significance of the mean and standard deviation in a probability distribution?
4. How would the distribution change if the sample size \( n \) were larger?
5. Can you explain why the distribution is right-skewed in this case?

**Tip:** Binomial distributions are symmetric when \( p = 0.5 \), but become more skewed as \( p \) moves away from 0.5.

USA Today reported that about 20% of all people in the United States are illiterate. Suppose you take eight people at random off a city street. Make a histogram showing the probability distribution of the number of illiterate people out of the eight people in the sample. Find the mean and standard deviation of this probability distribution.

Learn how to solve a binomial distribution problem involving the number of illiterate people in a random sample of eight, where 20% of the population is illiterate. This problem covers key probability concepts, including calculating the mean and standard deviation. Suitable for Grades 10-12.

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