Question

in a large population, 52% 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 not - vaccinated

The probability that a person is not vaccinated is $1 - 0.52=0.48$.

Step2: Find the probability that none of the 5 people is vaccinated

Since the selections are independent events, the probability that none of the 5 people is vaccinated is $(0.48)^5$.
$(0.48)^5=0.48\times0.48\times0.48\times0.48\times0.48 = 0.0254803968$.

Step3: Find the probability that at least one is vaccinated

The probability that at least one person is vaccinated is the complement of the event that none of them is vaccinated. Let $P(X\geq1)$ be the probability that at least one is vaccinated and $P(X = 0)$ be the probability that none is vaccinated. Then $P(X\geq1)=1 - P(X = 0)$.
$P(X\geq1)=1-(0.48)^5=1 - 0.0254803968 = 0.9745196032\approx0.9745$.

Answer:

$0.9745$