Twenty percent of U.S. residents are in their thirties. Consider a group of eight U.S. residents selected at random. Find the probability that two or three of the people in the group are in their thirties.
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Part 1
The probability that two or three of the people in the group are in their thirties is 
  
enter your response here.
​(Round to four decimal places as​ needed.)

To find the probability that two or three people out of eight randomly selected U.S. residents are in their thirties, given that the probability of any single person being in their thirties is 20% (or \( p = 0.20 \)), we can use the **binomial probability formula**.

The binomial probability formula is:
\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]
where:
- \( n \) is the total number of trials (in this case, 8),
- \( k \) is the number of successful outcomes (people in their thirties),
- \( p \) is the probability of success on a single trial (0.20),
- \( \binom{n}{k} \) is the number of combinations, calculated as \( \frac{n!}{k!(n - k)!} \).

We need to calculate the probability for exactly two people and exactly three people being in their thirties, then add these probabilities.

### Step 1: Calculate \( P(X = 2) \)
\[
P(X = 2) = \binom{8}{2} (0.20)^2 (0.80)^6
\]

1. \( \binom{8}{2} = \frac{8!}{2!(8 - 2)!} = 28 \)
2. Calculate \( (0.20)^2 = 0.04 \)
3. Calculate \( (0.80)^6 \approx 0.2621 \)

So,
\[
P(X = 2) = 28 \times 0.04 \times 0.2621 \approx 0.2932
\]

### Step 2: Calculate \( P(X = 3) \)
\[
P(X = 3) = \binom{8}{3} (0.20)^3 (0.80)^5
\]

1. \( \binom{8}{3} = \frac{8!}{3!(8 - 3)!} = 56 \)
2. Calculate \( (0.20)^3 = 0.008 \)
3. Calculate \( (0.80)^5 \approx 0.3277 \)

So,
\[
P(X = 3) = 56 \times 0.008 \times 0.3277 \approx 0.1465
\]

### Step 3: Add the Probabilities
Now, we add \( P(X = 2) \) and \( P(X = 3) \):
\[
P(X = 2 \text{ or } X = 3) = 0.2932 + 0.1465 = 0.4397
\]

### Final Answer
The probability that two or three of the people in the group are in their thirties is approximately:
\[
\boxed{0.4397}
\]

Would you like further explanation on any of these steps?

---

1. What is a binomial probability, and when is it useful?
2. How does the binomial formula change for different group sizes or probabilities?
3. Why do we add probabilities for "two or three" instead of calculating separately?
4. What is a combination, and how does it factor into the probability calculation?
5. How would this probability change if the group size increased?

**Tip:** When calculating binomial probabilities, keep track of each step to avoid errors in intermediate values.

Twenty percent of U.S. residents are in their thirties. Consider a group of eight U.S. residents selected at random. Find the probability that two or three of the people in the group are in their thirties.

Learn how to calculate the probability that two or three out of eight randomly selected U.S. residents are in their thirties. This solution involves binomial probability calculations with a 20% likelihood of selection, suitable for high school students studying advanced probability concepts.

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