A local county has an unemployment rate of 5.6%. A random sample of 15 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 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 1 in the sample are unemployed.

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

This is a binomial probability problem since each individual in the sample either is unemployed (with probability \( p = 0.056 \)) or is employed (with probability \( 1 - p \)).

For a binomial distribution, the probability of \( k \) unemployed people out of \( n \) (15 in this case) can be calculated as:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

where:
- \( n = 15 \)
- \( k \) is the number of unemployed people in the sample
- \( p = 0.056 \)

Let's solve each part.

### Part (a): Find the probability that exactly 4 in the sample are unemployed.

We are asked to find \( P(X = 4) \).

\[
P(X = 4) = \binom{15}{4} (0.056)^4 (1 - 0.056)^{15 - 4}
\]

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

This is the probability of \( X < 3 \), which means summing \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \):

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

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

This is the probability of \( X > 1 \), which can be calculated as:

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

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

This is the probability of \( X \leq 3 \), which means summing \( P(X = 0) \), \( P(X = 1) \), \( P(X = 2) \), and \( P(X = 3) \):

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

Let’s compute each probability using these formulas.Here are the probabilities for each part, rounded to four decimal places:

a) The probability that exactly 4 in the sample are unemployed: **0.0071**

b) The probability that there are fewer than 3 in the sample who are unemployed: **0.9518**

c) The probability that there are more than 1 in the sample who are unemployed: **0.2038**

d) The probability that there are at most 3 in the sample who are unemployed: **0.9918**

Would you like further explanation on any part of the solution?

Here are some related questions to explore further:
1. How would the probabilities change if the unemployment rate increased to 10%?
2. What is the expected number of unemployed people in a sample of 15?
3. How would the results vary with a larger sample size, say 30 people?
4. What if we wanted the probability of finding exactly 5 unemployed individuals in this sample?
5. How would we interpret the cumulative probability for each scenario?

**Tip:** Binomial probabilities are especially useful when dealing with situations that involve two possible outcomes—often framed as "success" and "failure."

Calculate probabilities for unemployment scenarios using a binomial distribution with a sample of 15 and a local unemployment rate of 5.6%. Find probabilities for exactly 4, fewer than 3, more than 1, and at most 3 unemployed individuals in the sample. Suitable for high school students.

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