Math 141: Probability Homework

Due Monday, October 13th

Problem 1

Students in a prior semester of Math 141 were asked two questions as part of a larger survey: “Are hot dogs sandwiches?” and “If dogs wore pants, would they wear them on their front legs, back legs, or on all four legs”? A summary of responses for 70 students is given below:

Hotdog is a sandwich Hotdog is not a sandwich
Front legs 1 1
Back legs 17 26
All 4 legs 9 16

Suppose we randomly choose 1 student who completed this survey.

  1. Are the events “the student thinks a hotdog is not a sandwich” and “the student thinks dogs should wear pants on their back legs” mutually exclusive?

  2. What is the probability that the randomly chosen student thinks a hotdog is a sandwich?

  3. What is the probability that the randomly chosen student thinks a hotdog is a sandwich and thinks that dogs should wear pants on all 4 legs?

  4. What is the probability that the randomly chosen student thinks a hotdog is a sandwich given that the student thinks dogs should wear pants on all 4 legs?

  5. Is the event “the student thinks a hotdog is a sandwich” independent (in the statistical sense) of the event “the student thinks dogs should wear pants on their front legs”? Explain your reasoning.

Problem 2

Suppose A and B are events with P(A) = 0.3 and P(B) = 0.7.

  1. Is it always possible to compute P(A and B), if you only know P(A) and P(B) and nothing else?

  2. Assuming that events A and B arise from independent random processes, find P(A and B) and P(A | B).

  3. Suppose we are given that P(A and B) = 0.1. Are events A and B independent? Explain.

  4. If we are given that P(A and B) = 0.1 , what is P(A | B)?

Problem 3

Lupus is a medical phenomenon where antibodies that are supposed to attack foreign cells to prevent infections instead see plasma proteins as foreign bodies, leading to a high risk of blood clotting. It is believed that 2% of the population suffer from this disease. The test is 98% accurate if a person actually has the disease. The test is 74% accurate if a person does not have the disease. There is a line from the Fox television show House that is often used after a patient tests positive for lupus: “It’s never lupus.” Do you think there is truth to this statement? Use appropriate probabilities to support your answer. (Hint: Bayes’ Law will be helpful!)