Question

people with type o - negative blood are universal donors. any patient can receive a transfusion of o - negative blood. only 7.2% of the american population has o - negative blood. if we choose 10 americans at random, what is the probability that at least 1 of them has o - negative blood? (round to 3 decimal places. leave your answer in decimal form.)

Explanation:

Step1: Calculate the probability of a person not having O - negative blood

The probability of a person having O - negative blood is \(p = 0.072\). So the probability of a person not having O - negative blood is \(q=1 - p=1 - 0.072 = 0.928\)

Step2: Calculate the probability that none of the 10 people have O - negative blood

Using the binomial probability formula for \(n = 10\) trials and \(k = 0\) successes (where a "success" is having O - negative blood), \(P(X = 0)=\binom{10}{0}p^{0}q^{10}\)
Since \(\binom{10}{0}=\frac{10!}{0!(10 - 0)!}=1\) and \(p^{0}=1\), then \(P(X = 0)=(0.928)^{10}\)
\((0.928)^{10}\approx0.478\)

Step3: Calculate the probability that at least 1 of the 10 people has O - negative blood

Using the complement rule \(P(X\geq1)=1 - P(X = 0)\)
\(P(X\geq1)=1-(0.928)^{10}\)
\(1 - 0.478 = 0.522\)

Answer:

\(0.522\)