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.