in a large population, 66% of the people have been vaccinated. if 5 people are randomly selected, what is…

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. question help: message instructor

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. question help: message instructor

Answer

Explanation:

Step1: Find the probability of a person not being vaccinated

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

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

Since the selections are independent events, the probability that none of the 5 people are vaccinated is $q^5$. Substituting $q = 0.34$ into the formula, we get $(0.34)^5=0.34\times0.34\times0.34\times0.34\times0.34 = 0.0045435424$.

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 $P(X\geq1)=1 - P(X = 0)$. Since $P(X = 0)=(0.34)^5$, we have $P(X\geq1)=1-(0.34)^5=1 - 0.0045435424 = 0.9954564576$.

Answer:

$0.9955$