there are four blood types, and not all are equally likely to be in blood banks. in a certain blood bank…
there are four blood types, and not all are equally likely to be in blood banks. in a certain blood bank, 49% of donations are type o blood, 27% of donations are type a blood, 20% of donations are type b blood, and 4% of donations are type ab blood. a person with type b blood can safely receive blood transfusions of type o and type b blood. what is the probability that the 4th donation selected at random can be safely used in a blood transfusion on someone with type b blood? (0.31)^3(0.69) (0.51)^3(0.49) (0.69)^3(0.31) (0.80)^3(0.20)
Answer
Explanation:
Step1: Calculate probability of non - usable blood
The probability that a donation is not safe for a Type B recipient is the probability of it being Type A or Type AB. So, $P(\text{not safe})=0.27 + 0.04=0.31$.
Step2: Calculate probability of usable blood
The probability that a donation is safe for a Type B recipient is the probability of it being Type O or Type B. So, $P(\text{safe})=0.49+0.20 = 0.69$.
Step3: Use the geometric - probability formula
We want the 4th donation to be the first safe one for a Type B recipient. The geometric - probability formula for the $k$th trial to be the first success is $P(X = k)=(1 - p)^{k - 1}p$, where $p$ is the probability of success on a single trial and $k$ is the trial number. Here, $k = 4$ and $p=0.69$, and $1 - p = 0.31$. So the probability is $(0.31)^{3}(0.69)$.
Answer:
$(0.31)^{3}(0.69)$