In a large population, 69% of the people have been vaccinated. If 5 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?

Give your answer as a decimal to 4 places. 
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We can solve this problem by first finding the probability that none of the 5 people selected have been vaccinated, and then subtracting this from 1 to get the probability that at least one person has been vaccinated.

### Steps:

1. **Probability that a person is not vaccinated**:  
   If 69% of people are vaccinated, the probability that a person is **not** vaccinated is:
   \[
   P(\text{not vaccinated}) = 1 - 0.69 = 0.31
   \]

2. **Probability that none of the 5 people are vaccinated**:  
   Since the selections are independent, the probability that all 5 people are not vaccinated is:
   \[
   P(\text{none vaccinated}) = 0.31^5
   \]

3. **Probability that at least one person is vaccinated**:  
   The probability that at least one person is vaccinated is the complement of the probability that none are vaccinated:
   \[
   P(\text{at least one vaccinated}) = 1 - P(\text{none vaccinated})
   \]

Now, let's calculate this:

\[
P(\text{none vaccinated}) = 0.31^5
\]
\[
P(\text{at least one vaccinated}) = 1 - 0.31^5
\]

I'll calculate the exact value.The probability that at least one of the 5 randomly selected people has been vaccinated is approximately **0.9971** (to four decimal places).

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the probability change if 3 people were randomly selected instead of 5?
2. What if the vaccination rate were 80% instead of 69%?
3. How can this problem be generalized for \( n \) people and a vaccination rate \( p \)?
4. What is the probability that exactly one person out of the 5 is vaccinated?
5. How does the assumption of independence affect this calculation?

**Tip:** Always use complements when calculating "at least one" probabilities, as it's often simpler.

In a large population, 69% of the people have been vaccinated. If 5 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated? Give your answer as a decimal to 4 places.

Find out how to calculate the probability that at least one out of 5 randomly selected people is vaccinated, given a 69% vaccination rate. This detailed step-by-step solution covers core probability concepts using the complement rule. Suitable for Grades 9-12.

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