A local county has an unemployment rate of 6.4%. A random sample of 18 employable people are picked at random from the county and are asked if they are employed. The distribution is a binomial. Round answers to 4 decimal places.

a) Find the probability that exactly 4 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 2 in the sample are unemployed.

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

Given the problem, we can define the parameters of the **binomial distribution** as follows:

- The number of trials \( n = 18 \) (since we are sampling 18 people).
- The probability of success (in this case, being unemployed) is \( p = 0.064 \) (the unemployment rate is 6.4%).

The **binomial distribution** formula is:
\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]
where:
- \( X \) is the number of unemployed people in the sample.
- \( k \) is the specific number of unemployed people we are calculating the probability for.
- \( \binom{n}{k} \) is the binomial coefficient.

### a) Find the probability that exactly 4 in the sample are unemployed.
For this, we are solving \( P(X = 4) \).

\[
P(X = 4) = \binom{18}{4} (0.064)^4 (1 - 0.064)^{18 - 4}
\]

I'll calculate this probability.The probability that exactly 4 people in the sample are unemployed is approximately \( P(X = 4) = 0.0203 \).

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

We need to find \( P(X < 3) \), which is the sum of the probabilities for \( X = 0 \), \( X = 1 \), and \( X = 2 \).

\[
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
\]

I'll calculate this next.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, you can calculate the probability for part b using the binomial distribution formula I provided earlier, or wait until the tool is available again for further calculations.

If you have any other questions or need more help with a different part of the problem, feel free to ask!

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Here are some related questions that might expand your understanding:

1. What is the expected value and variance for this binomial distribution?
2. How would you calculate the cumulative probability for \( P(X \leq 2) \) directly?
3. How does the binomial distribution change when the sample size increases?
4. What is the significance of using a binomial distribution in this context?
5. How does a small unemployment rate affect the distribution shape?

**Tip:** When working with binomial probabilities, using a calculator or software for summing multiple probabilities can save time!

A local county has an unemployment rate of 6.4%. A random sample of 18 employable people are picked at random from the county and are asked if they are employed. The distribution is a binomial. Round answers to 4 decimal places. a) Find the probability that exactly 4 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 2 in the sample are unemployed. d) Find the probability that there are at most 3 in the sample are unemployed.

This problem explores binomial probability calculations for a sample of 18 people with an unemployment rate of 6.4%. Learn how to find the probability of exactly 4 unemployed, fewer than 3 unemployed, more than 2 unemployed, and at most 3 unemployed using the binomial distribution. Includes step-by-step solutions and key probability concepts.

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