The unemployment rate in a city is 13%. If 10
people from the city are sampled at random, find the probability that at most 1
of them is unemployed.

Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.
(If necessary, consult a list of formulas.)

To solve this problem, we'll use the binomial distribution formula, as we're dealing with a fixed number of independent trials, each with two possible outcomes (employed or unemployed).

### Step 1: Define the parameters
- **n** (number of trials): 10 (since 10 people are sampled)
- **p** (probability of success, where "success" means being unemployed): 0.13
- **q** (probability of failure, where "failure" means being employed): \( q = 1 - p = 1 - 0.13 = 0.87 \)
- **k**: the number of unemployed people (we are interested in \( k \leq 1 \)).

### Step 2: Apply the binomial probability formula
The binomial probability formula is:
\[
P(X = k) = \binom{n}{k} p^k q^{n-k}
\]
Where:
- \( \binom{n}{k} \) is the binomial coefficient, which is calculated as \( \frac{n!}{k!(n-k)!} \).

### Step 3: Compute the probability for \( k = 0 \) and \( k = 1 \)

1. **For \( k = 0 \) (none are unemployed):**
   \[
   P(X = 0) = \binom{10}{0} \times 0.13^0 \times 0.87^{10} = 1 \times 1 \times 0.87^{10}
   \]
   \[
   P(X = 0) = 0.2634 \quad (\text{rounded to four decimal places})
   \]

2. **For \( k = 1 \) (exactly 1 is unemployed):**
   \[
   P(X = 1) = \binom{10}{1} \times 0.13^1 \times 0.87^{9} = 10 \times 0.13 \times 0.87^9
   \]
   \[
   P(X = 1) = 0.3958 \quad (\text{rounded to four decimal places})
   \]

### Step 4: Find the probability that at most 1 person is unemployed
To find the probability that at most 1 person is unemployed, sum the probabilities for \( k = 0 \) and \( k = 1 \):
\[
P(X \leq 1) = P(X = 0) + P(X = 1)
\]
\[
P(X \leq 1) = 0.2634 + 0.3958 = 0.6592
\]

### Final Answer
The probability that at most 1 of the 10 people sampled is unemployed is **0.66** (rounded to two decimal places).

Would you like more details or have any questions?

Here are 5 related questions you might find useful:
1. How does changing the sample size affect the probability calculation?
2. What is the probability that exactly 2 people are unemployed in the sample?
3. How does the unemployment rate influence the distribution of the number of unemployed people in the sample?
4. What is the expected number of unemployed people in the sample?
5. How would the result change if the unemployment rate was 20% instead of 13%?

**Tip:** Understanding how to apply the binomial distribution formula is crucial for solving problems related to random sampling with binary outcomes.

Calculate the probability that at most 1 person out of a sample of 10 is unemployed using the binomial distribution formula. Understand the application of probability theory with fixed number of trials and two possible outcomes (employed or unemployed).

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