Recall from Problem 3.11 that a patient with Type O+ blood can receive either Type O+ or Type O- blood, while a patient with Type O- blood can only receive Type O- blood. According to data collected from blood donors, 0.37 of white, non-Hispanic donors are Type \(\mathrm{O}+\) and 0.08 are Type O-. For the following questions, assume that only white, non-Hispanic donors are being tested.

(a) On average, how many donors would need to be randomly sampled for a Type O+ donor to be identified? With what standard deviation?

(b) What is the probability that 4 donors must be sampled to identify a Type O+ donor?

(c) What is the probability that more than 4 donors must be sampled to identify a Type O+ donor?

(d) What is the probability of the first Type O- donor being found within the first 4 people?

(e) On average, how many donors would need to be randomly sampled for a Type O- donor to be identified? With what standard deviation?

(f) What is the probability that fewer than 4 donors must be tested before a Type O- donor is found?

Data From Problem 3.11

When patients receive blood transfusions, it is critical that the blood type of the donor is compatible with the patients, or else an immune system response will be triggered. For example, a patient with Type O- blood can only receive Type O- blood, but a patient with Type O+ blood can receive either Type O+ or Type O-. Furthermore, if a blood donor and recipient are of the same ethnic background, the chance of an adverse reaction may be reduced. According to a 10-year donor database, 0.37 of white, non-Hispanic donors are O+ and 0.08 are O-. (a) Consider a random sample of 15 white, non-Hispanic donors. Calculate the expected value of individuals who could be a donor to a patient with Type O+ blood. With what standard deviation? (b) What is the probability that 3 or more of the people in this sample could donate blood to a patient with Type O-blood?