A local county has an unemployment rate of 7%. A random sample of 19 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 2 in the sample are unemployed. 

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

The situation described can be modeled using the **binomial distribution**, where each person has a probability of being unemployed (success) of \( p = 0.07 \), and there are 19 independent trials (since the sample size is 19).

The probability mass function (PMF) for a binomial distribution is given by:

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

Where:
- \( n = 19 \) (number of trials),
- \( p = 0.07 \) (probability of being unemployed),
- \( k \) is the number of unemployed people you're looking for,
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n - k)!} \).

We will use this formula for each part of the problem.

### a) Find the probability that exactly 2 in the sample are unemployed.
For this part, \( k = 2 \).

\[
P(X = 2) = \binom{19}{2} (0.07)^2 (1 - 0.07)^{19 - 2}
\]

### b) Find the probability that there are fewer than 3 in the sample unemployed.
Here we need to find \( P(X < 3) \), which is the sum of the probabilities for \( X = 0 \), \( X = 1 \), and \( X = 2 \).

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

### c) Find the probability that there are more than 2 in the sample unemployed.
Here we need to find \( P(X > 2) \), which is:

\[
P(X > 2) = 1 - P(X \leq 2)
\]

Since \( P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \), we can calculate it and then subtract from 1.

### d) Find the probability that there are at most 3 in the sample unemployed.
This is \( P(X \leq 3) \), which is the sum of the probabilities for \( X = 0 \), \( X = 1 \), \( X = 2 \), and \( X = 3 \).

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

Let me calculate these probabilities now.Here are the results for each part, rounded to 4 decimal places:

### a) Probability that exactly 2 in the sample are unemployed:
\[
P(X = 2) = 0.2440
\]

### b) Probability that fewer than 3 in the sample are unemployed:
\[
P(X < 3) = 0.8561
\]

### c) Probability that more than 2 in the sample are unemployed:
\[
P(X > 2) = 0.1439
\]

### d) Probability that at most 3 in the sample are unemployed:
\[
P(X \leq 3) = 0.9602
\]

Would you like more details on any of these calculations?

Here are five related questions you might find interesting:
1. What is the probability that no one in the sample is unemployed?
2. How would the probabilities change if the unemployment rate increased to 10%?
3. What is the expected number of unemployed people in the sample?
4. How would the probabilities change if the sample size increased to 30?
5. What is the standard deviation of the number of unemployed people in the sample?

**Tip**: The binomial distribution is useful for modeling situations where you have a fixed number of trials and two possible outcomes (success/failure).

A local county has an unemployment rate of 7%. A random sample of 19 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 2 in the sample are unemployed.

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

Solve a binomial probability problem where a sample of 19 people is drawn from a population with a 7% unemployment rate. Learn how to calculate the probabilities for exactly 2 unemployed people, fewer than 3, more than 2, and at most 3 unemployed people using the binomial distribution formula. Suitable for Grades 10-12.

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