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Solutions to Probability Questions

## Question #1: False Positive Rate of HIV Test

Question #1:

At one point in the 1980s, a widely used test to test for HIV (the disease that causes AIDS) had a false positive rate of about 7%. That is, if you gave the test to a bunch of people, even if NONE of them actually had the disease, the test would report a positive result in 7% of them. A physician in a small rural clinic administers this test about 5 times a week. What is the likelihood of finding at least one false positive case?

Question #2:

The 2000 census allowed each person to choose from a long list of races. That is, in the eyes of the Census Bureau, you belong to whatever race you say you belong to. If we choose a resident of the United States at random, the 2000 census gives these probabilities:

 Hispanic Not-Hispanic Asian 0.000 0.036 Black 0.003 0.121 White 0.060 0.691 Other 0.062 0.027

Let A be the event that a randomly chosen American identifies himself/herself as Hispanic. Let B be the event that the person identifies as white.

A.Verify that the table gives a legitimate assignment of probabilities.

B.Describe in words and find P(A)

C.Describe Bc in words and find P(Bc)

D.Express “the person chosen is a non-Hispanic white” in terms of events A and B. What is the probability of this event?

Question #3:

A national study indicates that 39% of Florida residents are foreign-born. Suppose that you randomly choose three Floridians so that each has probability 0.39 of being foreign-born and the three are independent of each other.  Let W be the number of foreign-born people you choose.

A.What are the possible values of W? That is, what is the sample space of W?

B.Looking at the three people in your sample, there are 8 possible arrangements of foreign (F) and domestic (D) birth. For example, FFD means the first two are foreign born and the third is not. List all 8 possible arrangements. Then provide the probability for each one.

C.Think back to the sample space for W in part ‘a’ above. For each of the 8 arrangements in part ‘b’ above, what is the value of W? For each possible value of W in the sample space, give its probability.

Question #4:

A.A statistics professor asks her graduate student to roll a die 10,000 times and record the results. Give the expected mean of the outcome.

B.The die roll experiment is repeated (though with a different graduate student – for some reason the previous one went to work with a different advisor). However in this case, the die is weighted so that a 6 shows up 30%, a 1 shows up 10% of the time and the remaining numbers (2,3,4,5) each show up 15% of the time. Now what is the mean of 10,000 rolls?

Question #5:

In a college population, students are classified by gender and whether or not they are frequent binge drinkers. Here are the probabilities:

 Men Women Binge Drinker 0.11 0.12 Not Binge Drinker 0.32 0.45

A.Find the probability that a randomly selected student is a male binge drinker, and find the probability that a randomly selected student is a female binge drinker.

B.Find the probability that a student is a binge drinker, given that the student is male and find the probability that a student is a binge drinker, given that the student is female. You can determine this with a simple calculation off of the chart, but you must confirm these values by using the conditional probabilities as discussed in lecture.

C.Your answer to part (a) gives a higher probability for females, while your answer for part (b) gives a higher probability for males. Interpret your answers in terms of the question of whether there are gender differences in binge-drinking behavior. Decide which comparison you prefer and explain the reasons for your preference.