n a large population, 66 % 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.

We are tasked with finding the probability that at least one of four randomly selected people from a population (where 66% are vaccinated) has been vaccinated.

### Approach:
It is easier to first calculate the complement: the probability that **none** of the four people are vaccinated, and then subtract that value from 1 to find the probability that at least one is vaccinated.

### Step 1: Probability that a person is **not** vaccinated:
Since 66% (or 0.66) of the population is vaccinated, the probability that a person is **not** vaccinated is:
\[
P(\text{not vaccinated}) = 1 - 0.66 = 0.34
\]

### Step 2: Probability that **none** of the 4 people are vaccinated:
For 4 people to be not vaccinated, we raise the probability of a single person not being vaccinated to the 4th power:
\[
P(\text{none vaccinated}) = (0.34)^4 = 0.34^4
\]

### Step 3: Complement (at least one vaccinated):
The probability that at least one person is vaccinated is the complement of none being vaccinated:
\[
P(\text{at least one vaccinated}) = 1 - P(\text{none vaccinated}) = 1 - 0.34^4
\]

Now let's compute \( 0.34^4 \) and subtract from 1.

The probability that at least one of the four randomly selected people has been vaccinated is approximately \( 0.987 \) (to three decimal places).

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the probability that exactly two out of the four selected people are vaccinated?
2. How would the result change if the probability of vaccination were 50% instead of 66%?
3. What is the probability that none of the four selected people are vaccinated?
4. How would the result differ if we selected 5 people instead of 4?
5. Can we calculate the probability of selecting at least two vaccinated people?

**Tip:** When solving probability problems, identifying the complement can often simplify calculations.

In a large population, 66% 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.

Find the probability that at least one out of four randomly selected people from a population where 66% are vaccinated has been vaccinated. This problem uses key concepts from probability theory, such as complementary probability and the complement rule. Suitable for students in Grades 9-12.

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