1. Suppose that at UVA, 76% of all undergraduates are in the College, 10% are in

Question

1. Suppose that at UVA, 76% of all undergraduates are in the College, 10% are in Engineering, 7% are in Commerce, 4% are in Nursing, and 3% are in Architecture. In each school, the percentage of females is as follows: 59% in the College, 23% in Engineering, 49% in Commerce, 88% in Nursing, and 33% in Architecture. If a randomly selected student is male, what is the probability that he's from the College?

2. Three airlines serve a small town in Ohio. Airline A has 48% of all scheduled flights, airline B has 29% and airline C has the remaining 23%. Their on-time rates are 83%, 64%, and 42%, respectively. A flight just left on-time. What is the probability that it was a flight of airline A?

3. Transplant operations have become routine. One common transplant operation is for kidneys. The most dangerous aspect of the procedure is the possibility that the body may reject the new organ. There are several new drugs available for such circumstances and the earlier the drug is administered, the higher the probability of averting rejection. The New England Journal of Medicine recently reported the development of a new urine test to detect early warning signs that the body is rejecting a transplanted kidney. However, like most other tests, the new test is not perfect. In fact, 20% of negative tests and 10% of positive tests prove to be incorrect. Physicians know that in about 29% of kidney transplants the body tries to reject the organ. If the new test has a positive result (indicating early warning of rejection), what is the probability that the body is attempting to reject the kidney?

4. You ask a neighbor to water a sickly plant while you are on vacation. Without water the plant will die with probability 0.9. With water it will die with probability 0.5. You are 83 % certain the neighbor will remember to water the plant. When you are on vacation:

(a) Find the probability that the plant will die.

(b) You come back from the vacation and the plant is dead. What is the probability the neighbor forgot to water it?

Explanation / Answer

Question 1:

Suppose that at UVA, 76% of all undergraduates are in the College, 10% are in Engineering, 7% are in Commerce, 4% are in Nursing, and 3% are in Architecture. In each school, the percentage of females is as follows: 59% in the College, 23% in Engineering, 49% in Commerce, 88% in Nursing, and 33% in Architecture. If a randomly selected student is male, what is the probability that he's from the College?

Ans. Here we are given that:

P(College ) = 0.76, P( Engineering ) = 0.1, P( commerce) = 0.07, P(Nursing ) = 0.04, P(Architecture ) = 0.03

Also we are given that P( Females | College ) = 0.59, P( Females | Engineering ) = 0.23, P( Females | Commerce ) = 0.49, P( Females | Nursing ) = 0.88 and P( Females | Architecture ) = 0.33

Now using addition law of probability that total probability of being a female is computed as:

P(Female) = P( Females | College )P(College) + P( Females | Engineering )P(Engineering) + P( Females | Commerce )P(Commerce) + P( Females | Nursing )P(Nursing) + P( Females | Architecture )P(Architecture)

P(Female) = 0.76*0.59 + 0.1*0.23 + 0.07*0.49 + 0.04*0.88 + 0.03*0.33 = 0.5508

Therefore, we get: P(Male ) = 1 - P(Female ) = 1 - 0.5508 = 0.4492

Now using Bayes theorem we have:

P( College | Male ) P(Male ) = P(Male | College ) P(College)

P(Male | College ) = 1- P(Female | College ) = 1 - 0.59 = 0.41

Also we know that : P(College ) = 0.76 and P(Male) = 0.4492

Therefore, we get:

P( College | Male ) *0.4492 = 0.41*0.76

P( College | Male ) = 0.41*0.76/ 0.4492 = 0.6937

Therefore 0.6937 is the required probability here.

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