In a large population, 52 % of the people have been vaccinated. If 4 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?

Give your answer as a decimal (to at least 3 places) or fraction

To find the probability that **at least one** of the 4 randomly selected people has been vaccinated, we can use the complement rule. The complement of "at least one has been vaccinated" is that **none** of the 4 people has been vaccinated.

### Steps:
1. **Define the probability that a person is vaccinated:**
   The probability that a randomly selected person is vaccinated is \( P(\text{vaccinated}) = 0.52 \).
   
2. **Find the probability that a person is NOT vaccinated:**
   The probability that a randomly selected person is not vaccinated is:
   \[
   P(\text{not vaccinated}) = 1 - P(\text{vaccinated}) = 1 - 0.52 = 0.48
   \]

3. **Find the probability that none of the 4 people is vaccinated:**
   Since the selections are independent, the probability that none of the 4 people is vaccinated is:
   \[
   P(\text{none vaccinated}) = (0.48)^4
   \]
   Now calculate \( (0.48)^4 \):
   \[
   (0.48)^4 = 0.0533
   \]

4. **Find the probability that at least one person is vaccinated:**
   The probability that **at least one** of the 4 people is vaccinated is the complement of the probability that none are vaccinated:
   \[
   P(\text{at least one vaccinated}) = 1 - P(\text{none vaccinated}) = 1 - 0.0533 = 0.9467
   \]

### Final Answer:
The probability that at least one of the 4 people has been vaccinated is **0.9467**.

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Would you like any further details or clarification on this? Here are some related questions for deeper understanding:

1. What is the probability that exactly one person out of the four is vaccinated?
2. How would the result change if the vaccination rate was 60% instead of 52%?
3. What is the probability that all four people are vaccinated?
4. How would the probability change if we randomly selected 10 people instead of 4?
5. What is the probability that none of the 4 people is vaccinated, given a vaccination rate of 30%?

**Tip:** When dealing with probabilities of multiple independent events, always consider the complement rule as it simplifies calculations in many scenarios.

In a large population, 52 % of the people have been vaccinated. If 4 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?

Learn how to calculate the probability that at least one person out of four randomly selected individuals has been vaccinated, given that 52% of the population is vaccinated. This solution uses the complement rule and probability of independent events. Suitable for students in Grades 9-12.

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