Write your name on the first page. Right beneath it write the pledge.
I pledge on my honor that I have not given or received any unauthorized assistance on this assignment/examination.
In doing so you agree to all the guidelines and instructions listed above in completing this assessment.
There are 5 numbered problems. [47 total points.] Budget about 1 minute per point? Do not spend undue time on any 2 – 3 point item.]
Please submit problems in order. (It is not necessary to submit one problem per page.)
1. [8] Assume that 3% of the people in a certain population have covid-19. Consider an antigen test to detect the virus. 98% of those who have the virus will test positive when administered this test. 85% of those of those in the population who do not have the virus will test negative.
a. [6] Consider a tested individual who tests positive. What is the probability this person has covid-19?
b. [2] Suppose the repeated tests on the same individual are independent. A person takes the test twice, testing positive both times. What is the probability this person has covid-19?
2. [10] There are 2 raspberry tootsie pops and 1 grape tootsie pop in a bag.
a. [6] You reach in the bag and pull a pop out at random. You then leave the kitchen to eat it. After you’ve left the kitchen, two pops of the same flavor as the one you’re eating are added to the bag. The process continues until you have taken (and eaten) k pops. What’s the probability that all the pops you eat are the same flavor?
b. [4] Given that all k pops are the same flavor, what’s the probability they are all raspberry? Simplify your result. Determine the limiting value of this probability as k →∞?
3. [12] Two coins are to be flipped. The first coin will land on heads with probability
p (0 ≤ p ≤ 1); the second with probability 1 – p. Assume that the results of the flips are independent. Let X equal the total number of heads that result.
a. [3] Find P(X = 1)
b. [3] Determine E[X].
c. [3] Determine Var(X).
d. [3] What value(s) of p maximize Var(X)? What value(s) minimize it? Support your answers with mathematics.
4. [8] Let X be a random variable with E[X] = 7 and Var[X] = 3. Find
a. [2] E[4 – 2X]
b. [4] V ar[4 – 2X]
c. [2] E[(4 – 2X)2]
5. [9] There are 5 red and 11 blue balls in a bucket. Balls are randomly selected one at a time without replacement.
a. [3] If 7 balls are selected, what is the probability that exactly 4 are blue?
b. [3] If 7 balls are selected, what is the probability the balls alternate colors?
c. [3] A run is defined as a sequence of balls all the same color. For instance, the sequence
B B R B R R R B B R contains 6 runs (they are underlined). If all 16 balls are randomly placed in sequence, what is the probability there are exactly 10 runs?