Multiplication of Probability Group 20 Group Members Au

Multiplication of Probability Group 20

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

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

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

Section 1: Revision Written by Aid Chan Lai Chun (98002770) Lam Po King (98003270) Recall the memory of definition probability Target To ensure each students has basic concept on probability

Section 2: Independent Events Written by Aid Chan Wing Kwan (98002930) Au Chun Kwok (98003350) Define what an independent event is Show examples and non-examples of independent events Target Students can differentiate what an independent event is

Section 3: Multiplication of Probability Written by Aid Chan Lai Chun (98002770) Chiu Wai Ming (98241940) Introduce 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

Section 4: Examples Written by Aid Chan Wing Kwan (98002930) Chiu Wai Ming (98241940) To 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

Section 5: Summary Written by Aid Chiu Wai Ming (98241940) Lam Po King (98003270) Revise and clarify some important concepts Target Help the students to consolidate the main ideas of this lesson

Section 6: Exercises Written by Aid Au Chun Kwok (98003350) Lam Po King (98003270) Show 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

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

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:

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

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

1. Revision I) Definition of Probability Example : The possible outcomes : 1 32 9 3 1 3 = ?

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

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

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

2. Independent Events Definition Two events are said to be independent of the happening of one event has no effect on the happening of the other. In ev depe en nd ts en t

2. Independent Events Example: 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. 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.

2. Independent Events Non-example: 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. Since the second number is greater than the first number, B depends on A. Therefore, A and B are NOT independent.

2. Independent Events Non-example: 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. Since a banana is chosen first and an apple is then chosen, D is affected by C. Therefore, C and D are NOT independent.

2. Independent Events Question 1 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. Are A and B independent?

2. Independent Events Congratulation! In ev depe en nd ts en t 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.

2. Independent Events Sorry. The correct answer is. . . In ev depe en nd ts en Throwing 2 dice. t 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.

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

2. Independent Events Sorry. The correct answer is. . . 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 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.

2. Independent Events Question 3 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. Are A and B independent?

2. Independent Events Congratulation! In ev depe en nd ts en t 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.

2. Independent Events Sorry. The correct answer is. . . In ev depe en nd ts en Throwing 1 die and 1 coin. t 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.

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)

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

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

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

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

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

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

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

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)

3. Multiplication of Probability and Con ? ! n o diti are s. B d t n n e a v e A t n e d n e p e ind

Example 1 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?

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 = 4 Total no. of girls = 5 P( Sam ) = P( Mary ) = Thus the probability is = =

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 English What is this? Mathematics

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? P(fail English) Thus the probability of Petermeans…… passes Chinese Peter pass Chinese and Mathematics only, that English and Mathematics only What is this? is

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

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

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.

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

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.

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

Exercise 2 My old alarm clock has a probability of 2/3 that it will go off. 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. 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. Find the probability that: (a) I get to work (b) I do not get to the bus stop in time.

Exercise 2 - Solution

Exercise 2 - Solution

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.

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)

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.

Exercise 3 - Solution

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?

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

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