1. COMP 10020 Lecturer:Dr. Saralees Nadarajah Office: Room 2.223, Alan Turing Building Email: mbbsssn2@machester.ac.ukmbbsssn2@machester.ac.uk

2. COMP 10020 Lectures 1. Mondays 10am, Humanities Bridgeford Cordingley 2. Thursdays 9am, Kilburn 1.1

3. COMP 10020 Example Classes 1. Mondays 4pm, LF15 2. Tuesdays 11am, LF15 3. Tuesdays 2pm, LF15 4. Fridays 12am, LF15

4. COMP 10020 Website http://www.maths.man.ac.uk/service/COMP10020/materials.php

5. COMP 10020 Requirements Co-requisites: None Pre-requisites: None Textbook: Not required

6. COMP 10020 Topics 1. Axioms of probability 2. Conditional probability and independence 3. Random variables 4. Some discrete distributions.

7. COMP 10020 Assessment 1. Two take-home assignments 2. Two questions in the final exam

8. Sets Sample Space

9. Sets E EcEc

10. E∩F is the overlap area (∩ =AND) E F

11. Sets EUF is the area Red or Yellow (U=OR) E F

12. Sets E∩F C is the area in Red E F

13. Sets E C ∩F is the area in Yellow F E

14. Sets E and F are mutually exclusive E F

15. Probability Pr (E) =

16. Example 1 A hat contains four slips numbered 1 to 4. Two drawn without replacement: Sample Space = ?

17. Example 2 Toss a coin: Sample Space = ? Pr(Head) = ? Pr(Tail) = ?

18. Example 3 Toss a 6-sided dice: Sample Space = ? Pr(“1” turned up) = ? Pr(“2” turned up) = ?

19. Example 4 A card picked from ordinary bridge deck: Pr(Card is ♠) = ? Pr(Card is King) = ?

20. Example 5 Toss three coins: Sample Space = ? Pr(No heads) = ? Pr(At least one head) = ?

21. Axioms of Probability 0 ≤ Pr(E) ≤ 1 Pr(Sample Space) = 1 Pr(EUF) = Pr(E) + Pr (F) if E and F are mutually exclusive

22. Additive Law Pr(EUF) = Pr(E) + Pr (F) – Pr(E∩F)

23. Complementary Law 1 Pr(E C ) = 1 - Pr(E)

24. Complementary Law 2 Pr(E C ∩F) = Pr(F) – Pr(E∩F)

25. Complementary Law 3 Pr(E∩F C ) = Pr(E) – Pr(E∩F)

26. Example 6 Given: Pr(E)=0.4, Pr(F)=0.3, Pr(E∩F)=0.2. Pr(EUF) = ? Pr(E C ) = ? Pr(F c ) = ?

27. Assignment 3 Due 2 December 2009 (Wednesday) Hand in to the Support Office

28. Example 7 A bowl contains slips numbered 1,2,…, 20. A slip drawn at random and its number noted: Sample Space = ? Pr(Number is Prime OR Divisible by 3) = ?

29. Example 8 The probability that a student passes mathematics is 2/3, and the probability that he passes biology is 4/9.If the probability of passing at least one course is 4/5, what is the probability that he will pass both courses?

30. Example 9 A bag contains 5 balls, 3 are red and 2 are yellow. Three balls are drawn without replacement. Describe the sample space.

31. Example 10 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).

32. Example 11 A six-sided die is loaded in a way that each even face is twice as likely as each odd face. All even faces are equally likely, as are all odd faces. For a single roll of this die find the probability that the outcome is less than 4.

33. Example 12 Is the following statement true: if A and B are mutually exclusive events then Pr (A ∩ B) = Pr (A) Pr (B). Justify your answer with a simple example.

34. Example 13 Anne and Bob each have a deck of playing cards. Each flips over a randomly selected card. Assume that all pairs of cards are equally likely to be drawn. Determine the following probabilities: (a) the probability that at least one card is an ace, (b) the probability that the two cards are of the same suit, (c) the probability that neither card is an ace.

35. Example 14 A die is rolled and a coin is tossed, find the probability that the die shows an odd number and the coin shows a head.

36. Example 15 A six--sided die is loaded in a way that each even face is twice as likely as each odd face. All even faces are equally likely, as are all odd faces. For a single roll of this die find the probability that the outcome is less than 4.

37. Example 16 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.

38. Example 17 Is the following statement true: if A and B are mutually exclusive events then Pr (A ∩ B) = Pr (A) Pr (B). Justify your answer with a simple example.

39. Example 18 If Pr (A) = 0.5 and Pr (B) = 0.4,but we have no further information about the events A and B, how big might Pr (A U B) be? How small might it be? How big might Pr (A ∩ B) be? How small might it be?

40. Example 19 Prove that for any two events A and B, we have Pr (A ∩ B)  Pr (A) + Pr (B) - 1.

41. Example 20 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).