1). 20 politicians are having a tea party, 6 Democrats and 14 Republicans. To prepare, they need to choose: 3 people to set the table, 2 people to boil the water, 6 people to make the scones. Each person can only do 1 task. (Note that this doesn't add up to 20. The rest of the people don't help.) (a) In how many different ways can they choose which people perform these tasks? (b) Suppose that the Democrats all hate tea. If they only give tea to 10 of the 20 people, what is the probability that they only give tea to Republicans? (c) If they only give tea to 10 of the 20 people, what is the probability that they give tea to 9 Republicans and 1 Democrat? 8. Let A and B be two events. Suppose the probability that neither A or B occurs is 2/3. What is the probability that one or both occur? 9. Let C and D be two events with P(C) = 0.25, P(D) = 0.45, and P(C D) = 0.1. What is P(Cc D)? 10. You roll a four-sided die 3 times. For this problem we'll use the sample space with 64 equally likely outcomes. (a) Write down this sample space in set notation. (b) List all the outcomes in each of the following events. (i) A = 'Exactly 2 of the 3 rolls are fours' (ii) B = 'At least 2 of the 3 rolls are fours' (iii) C = 'Exactly 1 of the second and third rolls is a 4' (iv) A C

2) Suppose we have 8 teams labeled T1, . . . , T8. Suppose they are ordered by placing their names in a hat and drawing the names out one at a time. (a) How many ways can it happen that all the odd numbered teams are in the odd numbered slots and all the even numbered teams are in the even numbered slots? (b) What is the probability of this happening? (Taken from the book by Dekking et. al. problem 4.9) The space shuttle has 6 O-rings (these were involved in the Challenger disaster). When launched at 81 F, each O-ring has a probability of failure of 0.0137 (independent of whether other O-rings fail). (a) What is the probability that during 23 launches no O-ring will fail, but that at least one O-ring will fail during the 24th launch of a space shuttle? (b) What is the probability that no O-ring fails during 24 launches?

3). Corrupted by their power, the judges running the popular game show America's Next Top Mathematician have been taking bribes from many of the contestants. Each episode, a given contestant is either allowed to stay on the show or is kicked off. If the contestant has been bribing the judges she will be allowed to stay with probability 1. If the contestant has not been bribing the judges, she will be allowed to stay with probability 1/3. Suppose that 1/4 of the contestants have been bribing the judges. The same contestants bribe the judges in both rounds, i.e., if a contestant bribes them in the first round, she bribes them in the second round too (and vice versa). (a) If you pick a random contestant who was allowed to stay during the first episode, what is the probability that she was bribing the judges? (b) If you pick a random contestant, what is the probability that she is allowed to stay during both of the first two episodes? Practice Exam 1: All Questions, Spring 2014 3 (c) If you pick random contestant who was allowed to stay during the first episode, what

is the probability that she gets kicked off during the second episode

4). Consider the Monty Hall problem. Let's label the door with the car behind it a and

the other two doors b and c. In the game the contestant chooses a door and then Monty

chooses a door, so we can label each outcome as 'contestant followed by Monty', e.g ab

means the contestant chose a and Monty chose b.

(a) Make a 3 3 probability table showing probabilities for all possible outcomes.

(b) Make a probability tree showing all possible outcomes.

(c) Suppose the contestant's strategy is to switch. List all the outcomes in the event 'the

contestant wins a car'. What is the probability the contestant wins?

(d) Redo part (c) with the strategy of not switching.

e). Two dice are rolled.

A = 'sum of two dice equals 3'

B = 'sum of two dice equals 7'

C = 'at least one of the dice shows a 1'

(a) What is P(A|C)?

(b) What is P(B|C)?

(c) Are A and C independent? What about B and C?

5) There is a screening test for prostate cancer that looks at the level of PSA (prostatespecific antigen) in the blood. There are a number of reasons besides prostate cancer that

a man can have elevated PSA levels. In addition, many types of prostate cancer develop

so slowly that that they are never a problem. Unfortunately there is currently no test

to distinguish the different types and using the test is controversial because it is hard to

quantify the accuracy rates and the harm done by false positives.

For this problem we'll call a positive test a true positive if it catches a dangerous type of prostate cancer. We'll assume the following numbers: Rate of prostate cancer among men over 50 = 0.0005 True positive rate for the test = 0.9 False positive rate for the test = 0.01