2. Multiplication of Probability Group 20

3. Group Members Au Chun Kwok(98003350) Chan Lai Chun(98002770) Chan Wing Kwan(98002930) Chiu Wai Ming(98241940) Lam Po King(98003270)

4. This power point file is aiding for teaching Probability. We advise students using it under teacher’s instructions. Hope you enjoy this package ! Before Use It’s not necessary to use this package step by step, you can use in any order as you like.

5. Contents 4 Examples 1 Revision 3 Multiplication of Probability 2 Independent Events 5 Summary 6 Exercises Some word from our group

6. Written byChan Lai Chun(98002770) Lam Po King(98003270) Section 1: Revision AidRecall the memory of definition probability TargetTo ensure each students has basic concept on probability

7. Written byChan Wing Kwan(98002930) Au Chun Kwok(98003350) Section 2: Independent Events AidDefine what an independent event is Show examples and non-examples of independent events Target Students can differentiate what an independent event is

8. Written by Chan Lai Chun(98002770) Chiu Wai Ming(98241940) Section 3: Multiplication of Probability AidIntroduce the multiplication law of probability by stating definition; also provide examples and non-examples Target Students can distinguish a problem whether it can apply multiplication law Students can apply the multiplication law to the problem correctly

9. Written byChan Wing Kwan(98002930) Chiu Wai Ming(98241940) Section 4: Examples AidTo show what can we use the multiplication method in the problems of probability Target Try to show and do the examples with the students Show the connection between multiplication method and the probability that students learned before

10. Written byChiu Wai Ming(98241940) Lam Po King(98003270) Section 5: Summary AidRevise and clarify some important concepts TargetHelp the students to consolidate the main ideas of this lesson

11. Written byAu Chun Kwok(98003350) Lam Po King(98003270) Section 6: Exercises AidShow some different types of example for the students Target Give a chance for the students to do some calculations on probability by applying multiplication law

12. Contents 4 Examples 1 Revision 3 Multiplication of Probability 2 Independent Events 5 Summary 6 Exercises Some word to say

13. 1. Revision I) Definition of Probability When all the possible outcomes of an event E are equally likely, the probability of the occurrence of E, often denoted by P(E), is defined as: I) Definition of Probability When all the possible outcomes of an event E are equally likely, the probability of the occurrence of E, often denoted by P(E), is defined as:

14. 1. Revision I) Definition of Probability Example : The possible outcomes : ? =

15. 1. Revision I) Definition of Probability Example : The possible outcomes : ? = 9 5 2 7 6 3 8 1 9 4

16. 1. Revision I) Definition of Probability Example : The possible outcomes : ? = 9 3 12 3 1 3

17. II) P(E)=1 the probability of an event that is certain to happen 1. Revision $5 Example :A coin is tossed. and Since the possible outcomes are 1 Why ? P(head or tail) = II) P(E)=1 the probability of an event that is certain to happen and they are also favourable outcomes.

18. Example : A die is thrown. III) P(E)=0 the probability of an event that is certain NOT to happen 1. Revision, and Why ? P(getting a ‘7’) = Since the possible outcomes are 0 III) P(E)=0 the probability of an event that is certain NOT to happen,,,

19. IV) 0 P(E) 1 0 1/2 1 1. Revision <<<< unlikely certain evenly likely impossible likely

20. Definition Two events are said to be independent of the happening of one event has no effect on the happening of the other. 2. Independent Events Independent events

21. 1.Throwing a die and a coin. Let A be the event that ‘1’ is being thrown. Let B be the event that ‘Tail’ is being thrown. Then A and B are independent events. 2. Independent Events 2. Choosing an apple and an egg. Let C be the event that a rotten apple is chosen. Let D be the event that a rotten egg is chosen. Then C and D are independent events. Example:

22. 1.Throwing 2 dice. Let A be the event that any number is thrown. Let B be the event that a number which is greater than the first number is thrown. Non-example: 2. Independent Events Since the second number is greater than the first number, B depends on A. Therefore, A and B are NOT independent.

23. 2.Choosing 2 fruits. Let C be the event that a banana is chosen first. Let D be the event that an apple is then chosen. Non-example: 2. Independent Events Since a banana is chosen first and an apple is then chosen, D is affected by C. Therefore, C and D are NOT independent.

24. Throwing 2 dice. Let A be the event that ‘Odd’ is being thrown. Let B be the event that ‘divisible by 3’ is being thrown. 2. Independent Events Question 1 Are A and B independent?

25. Throwing 2 dice. Event A = the result is ‘Odd’. Event B = the result is ‘divisible by 3’. Event A and event B are independent because they do not affect each other. Independent events 2. Independent Events Congratulation!

26. Throwing 2 dice. Event A = the result is ‘Odd’. Event B = the result is ‘divisible by 3’. Event A and event B are independent because they do not affect each other. Independent events 2. Independent Events Sorry. The correct answer is...

27. Catching 2 fishes. Let A be the event that the cat catches a golden fish first. Let B be the event that the cat then catches a fish which is NOT golden. 2. Independent Events Question 2 Are A and B independent?

28. Catching 2 fishes. Event A = the result is ‘Golden’. Event B = the result is ‘Not Golden’. After the first catching, the total number of outcomes and the number of favourable outcomes changes. That is, B depends on A; therefore, A and B are NOT independent. 2. Independent Events Sorry. The correct answer is...

29. 2. Independent Events Congratulation! Catching 2 fishes. Event A = the result is ‘Golden’. Event B = the result is ‘Not Golden’. After the first catching, the total number of outcomes and the number of favourable outcomes changes. That is, B depends on A; therefore, A and B are NOT independent.

30. Throwing 1 die and 1 coin. Let A be the event that ‘Odd’ is being thrown. Let B be the event that ‘Head’ is being thrown. 2. Independent Events Question 3 Are A and B independent?

31. Throwing 1 die and 1 coin. Event A = the result is ‘Odd’. Event B = the result is ‘Head’. Event A and event B are independent because they do not affect each other. Independent events 2. Independent Events Congratulation!

32. Throwing 1 die and 1 coin. Event A = the result is ‘Odd’. Event B = the result is ‘Head’. Event A and event B are independent because they do not affect each other. Independent events 2. Independent Events Sorry. The correct answer is...

33. 3. Multiplication of Probability If there are 2 independent events A and B, we can calculate the probability of A and B by applying to Multiplication Law: P(A and B) = P(A)  P(B)

34. What is the number of possible outcomes of tossing a coin ? What is the number of possible outcomes of throwing a die? 3. Multiplication of Probability

35. What is the number of possible outcomes of throwing a die? What is the number of possible outcomes of tossing a coin ? T H

36. 3. Multiplication of Probability What is the total number of possible outcomes of throwing a die and tossing a coin at the same time? T H

37. 3. Multiplication of Probability What is the total number of possible outcomes of throwing a die and tossing a coin at the same time? Total Possible Outcomes = 2  6 =12

38. 3. Multiplication of Probability What is the probability of getting a head and an odd number? P(H and Odd)

39. 3. Multiplication of Probability Try another one, OK!? GO... Try another one, OK!? GO... Try another one, OK!? GO...

40. 3. Multiplication of Probability Find the probability that one die shows an ‘Odd’ number and the other die shows a number ‘divisible by 3’.

41. 3. Multiplication of Probability Find the probability that one die shows an ‘Odd’ number and the other die shows a number ‘divisible by 3’. Solution: P(one odd and one divisible by 3) = P(one odd)  P(one divisible by 3)

42.  and 3. Multiplication of Probability A and B are independent events. Condition!?

43. In a class, there are 9 students, they are John, Peter, Paul, Sam, Mary, Anna, Susan, Sandy and Betty. What is the probability of choosing Sam and Mary as the monitor and monitress? Example 1

44. In a class, there are 9 students, they are John, Peter, Paul, Sam, Joe, Mary, Susan, Sandy and Betty. What is the probability of choosing Sam and Mary as the monitor and monitress? Total no. of boys = 4Total no. of girls = 5 P( Sam ) =P( Mary ) = Thus the probability is = =

45. Example 2 The probability of Peter to pass Chinese, English and Mathematics are 3/4, 4/5 and 1/3 respectively. Find the probability that he passes Chinese and Mathematics only? Chinese EnglishMathematics What is this?

46. The probability of Peter to pass Chinese, English and Mathematics are 3/4, 4/5 and 1/3 respectively. Find the probability that he passes Chinese and Mathematics only? Peter pass Chinese and Mathematics only, that means…… P(fail English)Thus the probability of Peter passes Chinese and Mathematics only is English What is this?

47. Example 3 One letter is chosen at random from each of the words: SELECTEDEFFECTIVEMETHOD Find the probability that three letters are the same.

48. Which letter should be choose : SELECTED, P(‘E’) = EFFECTIVE, P(‘E’) = METHOD, P(‘E’) = probability that three letters are the same SELECTED EFFECTIVE METHOD SELECTEDEFFECTIVEMETHOD E EFFECTIVEMETHODMETHOD

49. 5. Summary Definition of independent event : Two events are said to be independent of the happening of one event has no effect on the happening of the other. no effect independent

50. P(A and B) = P(A)  P(B) where A and B are independent events  and Condition: A and B are independent events independent events Then, how can we write?

51. Exercise 1 Euler goes to a restaurant to have a dinner. He can choose beef, pork or chicken as the main dish and the probability of each is 0.3, 0.2 and 0.5 respectively. He can choose rice, spaghetti or potato to serve with the main dish and the probability of each is 0.1, 0.6 and 0.3 respectively. Find the probability that he chooses beef with spaghetti.

52. Exercise 1- Solution Yeah! Finish! Let’s try example 2.

53. Exercise 2 My old alarm clock has a probability of 2/3 that it will go off. Find the probability that: (a) I get to work (b) I do not get to the bus stop in time. Even if it does go off, there is a probability of 1/6 that I’ll sleep through it and not get to the bus stop in time. If it doesn’t, there is still a probability of 3/4 that I’ll wake up anyway in time to catch the 8 o’clock bus.

54. Exercise 2 - Solution

56. Exercise 3 In a football match, team A has a penalty kick. The coach is deciding which player to take that place. It is known that the goalkeeper will defend the left, the central and the right parts with probabilities of 0.3, 0.2 and 0.5 respectively. (a) Find the probability that Beckham has a goal if the probability of kicking the ball to the left, central and right part is 0.6, 0.1 and 0.3.

57. Exercise 3 - Solution (a) P(Beckham has a goal) = 1- P(left of goalkeeper) x P(right of player) - P(central of goalkeeper) x P(central of player) - P(right of goalkeeper) x P(left of player)

58. Exercise 3 In a football match, team A has a penalty kick. The coach is deciding which player to take that place. It is known that the goalkeeper will defend the left, the central and the right parts with probabilities of 0.3, 0.2 and 0.5 respectively. (b) Find the probability that Owen has a goal if the probability of kicking the ball to the left, central and right part is 0.2, 0.5 and 0.3.

59. Exercise 3 - Solution

60. Exercise 3 In a football match, team A has a penalty kick. The coach is deciding which player to take that place. It is known that the goalkeeper will defend the left, the central and the right parts with probabilities of 0.3, 0.2 and 0.5 respectively. (c) Which player do you think the coach should choose?

61. Exercise 3 Ha ha… Coach should choose me (Owen) because my probability of having a goal is greater than yours.

62. Some interest web page related to probability: http://shazam.econ.ubc.ca/flip/index.html http://www.intergalact.com/threedoor/threedoor.cgi 1. A game flipping a number of coins 2. A game opening one of the three doors