In a large population, 56 % 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?


To solve this problem, we use the concept of complementary probability. The probability that at least one person out of the five selected has been vaccinated is the complement of the probability that none of the five have been vaccinated.

### Step 1: Define the probability that a person has not been vaccinated.
If 56% of the people have been vaccinated, then the probability that a person has not been vaccinated is:

\[
P(\text{Not Vaccinated}) = 1 - 0.56 = 0.44
\]

### Step 2: Calculate the probability that none of the 5 people have been vaccinated.
If none of the 5 people have been vaccinated, then each of them falls into the "not vaccinated" category. The probability that all 5 people are not vaccinated is:

\[
P(\text{None Vaccinated}) = 0.44^5
\]

Let's calculate this:

\[
0.44^5 \approx 0.0195
\]

### Step 3: Calculate the probability that at least one person has been vaccinated.
The probability that at least one of the 5 people has been vaccinated is the complement of the probability that none of them have been vaccinated:

\[
P(\text{At Least One Vaccinated}) = 1 - P(\text{None Vaccinated})
\]

\[
P(\text{At Least One Vaccinated}) = 1 - 0.0195 \approx 0.9805
\]

### Final Answer
The probability that at least one of the 5 selected people has been vaccinated is approximately **0.9805**.

Would you like more details or have any other questions?

Here are some related questions you might find helpful:
1. What is the probability that exactly 3 out of 5 selected people have been vaccinated?
2. What is the probability that none of the 5 selected people have been vaccinated?
3. If 10 people are selected, what is the probability that at least one has been vaccinated?
4. How does the probability change if 70% of the population is vaccinated?
5. What is the probability that at most 2 out of 5 selected people have been vaccinated?

**Tip:** Using complementary probability (i.e., calculating "at least one" by subtracting the probability of "none") is often easier and more efficient in scenarios like this.

Learn how to calculate the probability that at least one out of five randomly selected people has been vaccinated. This problem uses the concept of complementary probability and is suitable for advanced high school students.

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