1. Problem 1 The following facts are known regarding two events A and B: Pr(A∩B) = 0.2,Pr(AUB) = 0.6,Pr(A | B) = 0.5 Find the following: (i)Pr (A) (ii)Pr (B) (iii)Pr (B|A)

2. Problem 2 A box contains 5 floppy disks, 3 of which are defective. Three disks are drawn at random from this box without replacement. Let: A = at least 2 of the floppy disks that are drawn are defective B = at least 1 of the floppy disks that are drawn is defective. (i)What is Pr (A)? (ii)What is Pr (B)? (iii)What is Pr (A∩B)? (iv)Are A and B independent events?

3. Problem 3 A class in statistics contains 10 students, 3 of whom are 19, 4 are 20, and 3 are 21. Two students are selected at random without replacement from the class. Let X be the average age of the 2 selected students. Derive the probability mass function of X. Calculate E(X) and Var (X).

4. Problem 4 If X is a random variable with the PMF: p(x) = 1/5 for x = 1, 2, 3, 4, 5 find the following: (i) E (X) (ii) E (X 2 ) (iii) E (2 X ) (iv) Var (X)

5. Problem 5 Let A and B be events. Show that Pr (A ∩ B |B) = Pr (A|B), assuming that Pr (B) > 0.

6. Problem 6 A batch of one hundred items is inspected by testing two randomly selected items. If one of the two is defective, the batch is rejected. What is the probability that the batch is accepted if it contains five defectives?

7. Problem 7 Suppose that A and B are events such that Pr (A | B) = Pr (B | A) and Pr(A U B) = 1 and Pr(A ∩ B) > 0. Prove that Pr (A) > 1/2.

8. Problem 8 Prove that for any three events A, B, C, each having positive probability, Pr (A ∩ B ∩ C) = Pr (A) Pr (B | A) Pr (C | A ∩ B).

9. Problem 9 A town has two fire engines operating independently. The probability that a specific fire engine is available when need is 0.99. What is the probability that neither is available when needed? What is the probability that a fire engine is available when needed?

10. Problem 10 You throw two fair dice, one green and one red. Decide which of the following pairs of events are independent: A) Sum is 5 & Red die shows 2 B) Sum is 5 & Red die shows even C) Sum is 5 & Sum is 4 D) Sum is even & Red die shows even

11. Problem 11 A bag contains 5 balls, 3 are red and 2 are yellow. Three balls are drawn without replacement. What is the probability of drawing at least two red balls?

12. Problem 12 As accounts manager in your company, you classify 75% of your customers as “good credit” and the rest as “risky credit” depending on their credit rating. Customers in the “risky” category allow their accounts to go overdue 50% of the time on average, whereas those in the “good” category allow their accounts to become overdue only 10% of the time. What percentage of overdue accounts are held by customers in the “risky credit” category?

13. Problem 13 Out of the students in a class, 60% are geniuses, 70% love chocolate, and 40% fall into both categories. Determine the probability that a randomly selected student is neither a genius nor a chocolate lover.

14. Problem 14 Let C be the event “exactly one of the events A and B occurs.” Express Pr (C) in terms of Pr (A), Pr (B) and Pr (A ∩ B).