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The problem you're dealing with involves **binomial probability**. The unemployment rate is given as 6.3%, and a random sample of 18 employable people is taken. The questions ask about different probabilities based on this binomial distribution.

Key parameters:
- Probability of being unemployed, \( p = 0.063 \)
- Probability of being employed, \( q = 1 - p = 0.937 \)
- Number of trials (sample size), \( n = 18 \)

The binomial probability formula is:
\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]
Where:
- \( P(X = k) \) is the probability of exactly \( k \) unemployed people,
- \( \binom{n}{k} \) is the binomial coefficient \( \frac{n!}{k!(n-k)!} \).

### a) Find the probability that exactly 2 in the sample are unemployed.

We want \( P(X = 2) \), so we use the formula with \( k = 2 \):
\[
P(X = 2) = \binom{18}{2} \times (0.063)^2 \times (0.937)^{16}
\]

### b) Find the probability that there are fewer than 3 in the sample unemployed.

This is \( P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \).

### c) Find the probability that there are more than 1 in the sample unemployed.

This is \( P(X > 1) = 1 - [P(X = 0) + P(X = 1)] \).

### d) Find the probability that there are at most 3 in the sample unemployed.

This is \( P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \).

Let me calculate these probabilities for you.Here are the calculated probabilities:

- **a) The probability that exactly 2 in the sample are unemployed**: \( 0.2144 \)
- **b) The probability that fewer than 3 in the sample are unemployed**: \( 0.8995 \)
- **c) The probability that more than 1 in the sample are unemployed**: \( 0.3149 \)
- **d) The probability that at most 3 in the sample are unemployed**: \( 0.9764 \)

Would you like a detailed explanation of any of these steps or more questions based on this problem?

Here are 5 related questions to expand on this:
1. How does the binomial probability distribution change if the sample size is increased?
2. What would happen to these probabilities if the unemployment rate was 10% instead of 6.3%?
3. How would you calculate the expected number of unemployed individuals in the sample?
4. How does the probability change if we want exactly 1 unemployed person in a sample of 18?
5. Can you apply the cumulative binomial probability to determine \( P(X \leq 4) \)?

**Tip**: When using binomial distributions, \( n \) and \( p \) significantly affect the shape of the distribution and the resulting probabilities.

A local county has an unemployment rate of 6.3%. A random sample of 18 employable people are picked at random from the county and are asked if they are employed. Round answers to 4 decimal places.

a) Find the probability that exactly 2 in the sample are unemployed.

b) Find the probability that there are fewer than 3 in the sample are unemployed.

c) Find the probability that there are more than 1 in the sample are unemployed.

d) Find the probability that there are at most 3 in the sample are unemployed.

Explore how to calculate binomial probabilities for a sample of 18 people with a 6.3% unemployment rate. Learn to solve for exact, cumulative, and complement probabilities, including examples such as the probability of exactly 2 unemployed individuals, fewer than 3, more than 1, and at most 3 unemployed individuals. Suitable for high school and college-level statistics.

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