Table 2 Outcomes in the sample space

s1 s2 s3 s4 s5 s6 s7

Pr

.2

.3

.1

.05

.05

.1

.2

1

. A = {s1, s2}, B = {s2, s3, s4}, C = {s3, s4, s6}, D = {s2, s7,s6}, E = {s1, s7}

2

. Using Table 2, show your work as you determine whether A and D are independent

. 3

. Using Table 2, show your work as you determine whether A and F are independent

.

4

. Using Table 2, Calculate Pr[B | C] directly; calculate it again using Bayes’ Theorem

.

5

. What is the general formula for the conditional probability of event A occurring, given that event B occurs?

6

. Give the definition of statistical independence of two events

.7

. If Pr[A] =

.4, and Pr

** = .2, and A and B are independent, then what is Pr[A and B]?**

8. If Pr[A] = .4, and Pr** = .2, and A and B are independent, then what is Pr[A | B]?**

9. If Pr[A] = .4, and Pr** = .2, and A and B are independent, then what is Pr[B | A]?**

10. If Pr[A] = .4, and Pr** = .2, and A and B are mutually exclusive, then what is Pr[B | A]?**

11. Prove that if both A and B have non-zero probabilities, and if A is statistically **independent of B, then B is statistically independent of A. **

12. What does i.i.d. stand for?

13. What is a random sample, as defined in the slides?

A natural gas pipeline runs across a faultline somewhere in the desert. Near the faultline, the manufacturers installed several identical leak detectors to detect if a rupture in the pipe occurs. Each detector also operates independently from the others. Moreover, each detector has the same chance of a “false alarm” – i.e., indicating a leak when there is none present.

14. If there are 3 such detectors (each with a 20% chance of a false alarm), and they all indicate a leak, what is the probability that all of them are producing false alarms (which would mean there is no leak present)?

15. If each detector has a 40% chance of a false alarm, what is the minimum number of detectors that would need to be installed so that if they all go off, the probability that they are all producing false alarms is less than .01? Less than .001?

16. If there are 5 such detectors, and the probability that they all deliver a false alarm is .001, then what is the probability that any one of them produces a false alarm?

17. Draw a table that defines a sample space and assigns probabilities to each elementary outcome, and define two events A and B. Do the calculations to show that Pr(A&B) Pr(A|B)**.**

20. Draw a table that defines a single sample space and assigns probabilities to each elementary outcome, and define events A, B, and C. Do the calculations to show that (i) Pr(A) > Pr(A|B), (ii) Pr(A) > Pr(A|C), and (iii) Pr(A) Pr(H | E).

25. Pr(E) = Pr(H | E).

26. Pr(E) Pr(H | E).

28. Pr(E | H) = Pr(H | E).

29. Pr(E | H) < Pr(H | E).

30. Give a formula for Bayes' theorem (either form that we've seen will do).

31. Prove that Pr(A | B) = Pr(A)Pr(B | A)/Pr(B).

32. For any events A and B with nonzero probabilities, Pr(A | B) ? Pr(A & B); explain.

33. At the UCI SportBall center concession stand, 70% of the visitors buy drinks, and 50% of the visitors buy

49. You tell your friend that you have a 30% chance of getting your first-choice summer internship, and a 40% chance of getting your second-choice internship (the two internship offers are with different companies, and so are decided independently). What are your chances of getting at least one of these internships?

50. Suppose events A, B, C, and D are i.i.d., with a shared probability of .1. In a given trial, what is the probability that at least one of these events will occur? [Hint: Use the rules of probability, and consider the event: not A & not B & not C & not D; if this event does not occur, what must be the case?]

51. You've applied for 12 summer internships, and you guess that you have a 10% chance of getting any one of them (where each of your applications will be judged independently of the others. What are your chances of being offered at least one internship? [Hint: Use the rules of probability, and consider the probability that you won't be offered any internships at all.]