The unemployment rate in a city is $$ 12 \% $$. If 7 people from the city are sampled at random, find the probability that at most 2 of them are 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.)

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UpStudy AI · Accepted Answer

To find the probability that at most 2 out of 7 randomly sampled people are unemployed in a city with a 12% unemployment rate, we can use the **binomial probability formula**:

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

Where:
- $$ n = 7 $$ (number of trials)
- $$ k $$ is the number of successes (unemployed individuals)
- $$ p = 0.12 $$ (probability of success on a single trial)

We need to calculate $$ P(X \leq 2) $$, which is the sum of probabilities for $$ k = 0, 1, 2 $$.

### Step-by-Step Calculations:

1. **Calculate $$ P(X = 0) $$:**
   $$
   P(X = 0) = \binom{7}{0} (0.12)^0 (0.88)^7 = 1 \times 1 \times 0.88^7 \approx 0.4087
   $$

2. **Calculate $$ P(X = 1) $$:**
   $$
   P(X = 1) = \binom{7}{1} (0.12)^1 (0.88)^6 = 7 \times 0.12 \times 0.88^6 \approx 0.3901
   $$

3. **Calculate $$ P(X = 2) $$:**
   $$
   P(X = 2) = \binom{7}{2} (0.12)^2 (0.88)^5 = 21 \times 0.0144 \times 0.88^5 \approx 0.1596
   $$

4. **Sum the probabilities:**
   $$
   P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \approx 0.4087 + 0.3901 + 0.1596 = 0.9584
   $$

5. **Round to two decimal places:**
   $$
   \boxed{0.96}
   $$

### Final Answer:
$$\boxed{0.96}$$
The probability that at most 2 out of 7 randomly sampled people are unemployed is **0.96**.

Upstudy AI Solution

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Extra Insights

Step 1: Calculate each probability

Step 2: Sum up the probabilities

Step 3: Final round off

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