Question

in a large population, 66% 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.

Explanation:

Step1: Find the probability of a person not being vaccinated

The probability that a person is vaccinated is $p = 0.66$. So the probability that a person is not vaccinated is $q=1 - p=1 - 0.66 = 0.34$.

Step2: Find the probability that none of the 5 - selected people are vaccinated

Since the selections are independent events, the probability that none of the 5 people are vaccinated is $q^n$, where $n = 5$. So the probability that none of the 5 people are vaccinated is $(0.34)^5=0.34\times0.34\times0.34\times0.34\times0.34 = 0.0045435424$.

Step3: Find the probability that at least one person is vaccinated

The probability that at least one person is vaccinated is the complement of the event that none of the people are vaccinated. Let $P(X\geq1)$ be the probability that at least one person is vaccinated. Then $P(X\geq1)=1 - P(X = 0)$. So $P(X\geq1)=1-(0.34)^5=1 - 0.0045435424 = 0.9954564576\approx0.9955$.

Answer:

$0.9955$