People with type O-negative blood are universal donors whose blood can safely be

Question

People with type O-negative blood are universal donors whose blood can safely be given to anyone. Only 8.25% of the population has O-negative blood. A mobile blood center is visited by 20 donors in the afternoon. Let X denote the number of universal donors among them. You may assume that X is binomial. Answer the following. Please let me know your method of finding the probabilities and include as much information as possible, rounding your final answers to the nearest thousandth (three decimal places).

a) The probability that exactly 3 of the donors have O-negative blood.

b) The probability that at least 3 of the donors have O-negative blood.

c)The mean (expected value) and standard deviation of X, the number of universal donors out of 20. Interpret both measures in a full sentence within the context of the problem

Explanation / Answer

Here there are 20 donors in the afternoon. Here X denote the number of universal donors.

X is bionmial so

P(x) = 20Cx (0.0825)X (0.9175)20-X

(A) Here we have to calculate the probability of exactly 3 donors.

so P(3) = 20C3 (0.0825)3 (0.9175)20-3

= 0.1481

(B) Probability that at least 3 of the donors have O- negative blood. so that means that 3 or more people are having ) -negative blood out of 20 visitors.

so P(X>=3) = 1 - P(0) - P(1) - P(2)

here P(0) = no O- negativ4e blood donor, P(1) = 1 o- negative blood donor arrieved and P(2) = 2 O- negetiave blod donor arrived.

P(X>=3) =  1- [ 20C0 (0.0825)0 (0.9175)20 + 20C1 (0.0825)1 (0.9175)19 + 20C2 (0.0825)2 (0.9175)18]

= 1 - [ 0.1786 + 0.3214 + 0.2745]

= 0.2255

(c) The mean value of X = np = 20 * 0.0825 = 1.65

Standard deviation = sqrt (np(1-p) ) = sqrt (20 * 0.0825 * 0.9175) = 1.23

The expecnted value of O- negative blood in a given afternoon is 1.65 donors with deviation of 1.23 .

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