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Let's work through each part of this problem based on the probability distribution table provided:

### (a) Probability that one or more adults live with a disability:
The probability that **none** of the five adults live with a disability is given by \( P(x = 0) = 0.222 \). The probability that one or more adults live with a disability is the complement of this event:

\[
P(\text{one or more}) = 1 - P(x = 0)
\]
\[
P(\text{one or more}) = 1 - 0.222 = 0.778
\]

### (b) Probability that two or more adults live with a disability:
The probability that **two or more** adults live with a disability can be found by subtracting the probabilities for **zero** and **one** adult living with a disability from 1:

\[
P(\text{two or more}) = 1 - (P(x = 0) + P(x = 1))
\]
\[
P(\text{two or more}) = 1 - (0.222 + 0.390) = 1 - 0.612 = 0.388
\]

### (c) Probability that three or more adults live with a disability:
Similarly, the probability that **three or more** adults live with a disability can be found by subtracting the probabilities for **zero, one, and two** adults from 1:

\[
P(\text{three or more}) = 1 - (P(x = 0) + P(x = 1) + P(x = 2))
\]
\[
P(\text{three or more}) = 1 - (0.222 + 0.390 + 0.274) = 1 - 0.886 = 0.114
\]

### (d) Expected number of adults with a disability:
The expected value \( \mu \) is calculated using the formula for the mean of a probability distribution:

\[
\mu = \sum [x \cdot P(x)]
\]

Calculating step by step:

\[
\mu = (0 \times 0.222) + (1 \times 0.390) + (2 \times 0.274) + (3 \times 0.096) + (4 \times 0.017) + (5 \times 0.001)
\]
\[
\mu = 0 + 0.390 + 0.548 + 0.288 + 0.068 + 0.005 = 1.299
\]

### (e) Standard deviation of adults with a disability:
The standard deviation \( \sigma \) is calculated using the formula:

\[
\sigma = \sqrt{\sum [x^2 \cdot P(x)] - \mu^2}
\]

First, compute \( \sum [x^2 \cdot P(x)] \):

\[
\sum [x^2 \cdot P(x)] = (0^2 \times 0.222) + (1^2 \times 0.390) + (2^2 \times 0.274) + (3^2 \times 0.096) + (4^2 \times 0.017) + (5^2 \times 0.001)
\]
\[
\sum [x^2 \cdot P(x)] = 0 + 0.390 + 1.096 + 0.864 + 0.272 + 0.025 = 2.647
\]

Now, calculate the standard deviation:

\[
\sigma = \sqrt{2.647 - (1.299)^2} = \sqrt{2.647 - 1.687} = \sqrt{0.960} \approx 0.98
\]

### Final Answers:
- (a) Probability one or more: **0.778**
- (b) Probability two or more: **0.388**
- (c) Probability three or more: **0.114**
- (d) Expected number \( \mu \): **1.30**
- (e) Standard deviation \( \sigma \): **0.98**

Would you like further details or have any other questions?

Here are 5 related questions to explore:
1. What is the probability that exactly 2 adults have a disability?
2. How would the expected value change if the sample size increased to 10 adults?
3. What does the complement rule in probability represent, and how is it used here?
4. How is standard deviation interpreted in this context?
5. How could a change in the overall percentage of adults with a disability (from 26%) affect this distribution?

**Tip**: Always check that the total sum of probabilities in a probability distribution equals 1.

According to a certain organization, approximately 26% of adults in the United States live with a disability. This includes disabilities affecting mobility, cognition, vision, hearing, and others. Suppose that we select a random sample of five US adults, and let x = the number of adults out of five that live with a disability. The methods of Section 6.2 can be used to compute the probability assignments for the x distribution. Find the probabilities for parts (a) through (e).

Explore how to compute the probabilities of adults with disabilities in a random sample of five US adults. This problem involves key concepts such as binomial probability, expected value, and standard deviation, including formulas for calculating one or more events, two or more events, and more. Ideal for college-level students studying probability and statistics.

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