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. enter an integer or decimal number, accurate to at least 4 decimal places more... question help: message instructor

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 - selected people are vaccinated

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

Step3: Find the probability that at least one of the 5 people is vaccinated

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

Answer:

$0.9955$