Question

Universal blood donors People with type O-negative blood are universal donors. That is, any patient can receive a transfusion of O-negative blood. Only $7.2\%$, of the American population have O-negative blood. If $10$ people appear at random to give blood, what is the probability that at least $1$ of them is a universal donor? Follow the four-step process.

Short answer

P (at least one O) = $52.63\%$

Step-by-step solution

Step 1. Given Information

Because O-negative blood donors are universal, $7.2$ percent of the American population has O-negative blood.

Step 2. Concept used

Multiplication rule:$P(A\cap B)=P(A\cap B)\times P(B/A)$

Complement rule: $P(notA)=1-P\left(A\right)$

Step 3. Calculation

$P\left(O\right)=7.2\%=0.072$

Complement rule:

$P(notA)=1-P\left(A\right)$

To determine the likelihood of not having type O, use the complement rule:

$P(notO)=1-P\left(O\right)=1-0.072=0928$

Multiplication rule:

$P(A\cap B)=P\left(A\right)\times P\left(B\right)$

To find the likelihood of $10$people not possessing type O, use the multiplication rule:

$P(10notO)=(P(notO\left)\right)10=(0.928)10\approx 0.4737$

To calculate the likelihood of finding at least one individual with type O, use the complement rule:

$$\begin{gathered} P(atleastoneO)=1-P(10notO) \\ P(atleastoneO)=1-0.4737 \\ P(atleastoneO)=0.5263=52.63\% \end{gathered}$$