THE CHINESE UNIVERSITY OF HONG KONG

EDD 5161

Educational Communications and Technology

Group 2

Instructor: Dr. LEE FONG LOK

Group Members

NG TAT YEUNG (S98036770)

POON KIN MAN (S98115970)

Mathematics: Probabilities

Target audience: F5 student

Type of software: Lecturing

已知：鹵味與齋鹵味之和是三十六件，齋鹵味是鹵味的一點二倍，現隨機取一件，求… .

ContentsA. Revision

C. Complementary Events

E. Multiplication of Probability

F. Examples

G. Exercises

B. Mutually exclusive events

D. Independent Events

A. Revision

When all possible outcomes under consideration are equally likely to happen, then the probability of the happening of an event E, P(E) is given by:

P(E)=outcomes possible ofnumber Total

event the tofavourable outcomes ofNumber

(1) Definitions

Example 1:

(2) The possible outcomes

An unbiased coin

The total possible outcomes is head (H) or Tail (T)

P(H)=2

1and P(T)=

2

1

Example 2: A Fair die

P( ) =Odd numbers6

3

Total possible outcomes is 6

2

1

(3) Special Dice

What is the possible outcome of the above case?

Example 3:

There are 5 red marbles, 3 green marble and 2 black marbles

P( )=

105

2

1red marbles

An event(E) that is certain to happen, then

P(E) = 1

e.g. A die is thrown

(4) Certain and impossible

P(integers)= 6

6=1

(4) Certain and impossible

An event(E) that is impossible to happen, then

P(E) = 0

e.g. A die is thrown

P(getting a ‘0’) =6

0= 0

P(E) = 0P(E) = 0

0 P(E) 1

certainimpossible

(5) Conclusion

When the probability is greater than 0.5, implies the event is likely to happen

When the probability is smaller than 0.5, implies the event is unlikely to happen

0 P(E) 1 0 P(E) 1

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(B) Mutually exclusive eventsTwo events are said to be mutually exclusive events if both events cannot happen at the same time.

Example A die is rolled

Event A: getting a

Event B: getting a

Event C: getting a multiple of 3

Which are the mutually exclusive events?

Event A: getting a

Event B: getting a Correct

Answer: A and B

Addition of Probabilities

When two events E and F are mutually exclusive, then

P(E or F) = P(E) + P(F)

Example:If a card is drawn at random from a pack of 52 playing cards, find the probability that

Either a ‘king’ or a ‘queen’ is drawn

P(king or queen) = P(king) + P(queen)king queen

52

4+

52

4=

2

1

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(C) Complementary EventsGiven an event E, its complementary event E’ is the event that ‘E does not happen’. We have

P(E) + P(E’) = 1

Example:

P(A) + P(B) =1/2 + 1/2 = 1

Event A: getting a head

Event B: getting a Tail

Tossing a coinAre event A and B complementary ?

More example

Eventcomplementary

eventIn a Mathematics Test

Event A: will fail in the test

Event B: will pass in the test

Rolling a die

Event A: getting a ‘6’

Event B: getting an odd number

Drawing a card randomly from a pack of playing cards.

Event A: Getting a red card.

Event B: getting a ‘spade’

D. Independent Event

The occurrence of one event does not affect the probability of the occurrence of the other are simply called independent events.

Event A: Getting a head of a coin

Event B: Getting a ‘1’ of a die

A and B are independent events Maina

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E. Multiplication of Probability

For two independent events E and F,

P(E and F) = P(E) P(F)P(E and F) = P(E) P(F)P(E and F) = P(E) P(F)

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Complementary Events

P(E)+P(E’)=1P(E)=1-P(E’)P(E’)=1-P(E)

The probability that John will pass a test is .4

3

The probability that he will not pass the test is .4

1

4

31

F. Example

In a soccer match between two teams A and B, the probability that team A will win is 0.25 and probability that team B will win is 0.3. Find the probability that (a) team A or team B will win the match,(b) the two teams tie.

Example of Complementary Events

AnswerAnswer

(a) P(team A or team B wins) = P(team A wins) + P(team B wins)

(b) P(two teams tie) = 1- P(team A or team B wins) Complementary

Complementary

Events ?Events ?

= 0.25 + 0.3= 0.55

= 1-0.55= 0.45

Multiplication of Probability

P(E and F)=P(E) X P(F)P(E and F)=P(E) X P(F)

For two independentFor two independentevents E and F !events E and F !

Multiplication of Probability

P(E and F)P(E and F)=P(E) X P(F after E has occurred)=P(E) X P(F after E has occurred)

For two dependentFor two dependentevents E and F!events E and F!

Two cards are drawn one after the other at randomTwo cards are drawn one after the other at randomfrom a pack of 52 play cards. The first card drawn from a pack of 52 play cards. The first card drawn is put back into the pack and the pack is shuffledis put back into the pack and the pack is shuffledbefore the second card is drawn. before the second card is drawn. Find the probability thatFind the probability that(a) the first card drawn is a ‘king’ and the second (a) the first card drawn is a ‘king’ and the second card is a ‘club’,card is a ‘club’,(b) both cards drawn are the ‘king’ of clubs.(b) both cards drawn are the ‘king’ of clubs.

Example of Multiplication of Probabilities

(a) P(first king) (a) P(first king)

13

152

4

P(second club)P(second club)

4

152

13

P(first king and second club)P(first king and second club)=P(first king) X P(second club)=P(first king) X P(second club)

52

14

1

13

1

(b)(b) P(first king of clubs)P(first king of clubs)= P(second king of clubs)= P(second king of clubs)

52

1

P(both king of clubs)P(both king of clubs)=P(first king of clubs and second king of clubs )=P(first king of clubs and second king of clubs )=P(first king of clubs) X P(second king of clubs)=P(first king of clubs) X P(second king of clubs)

2704

152

1

52

1

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In a toys factory, two machines X and Y are used toIn a toys factory, two machines X and Y are used toproduce 70% and 30% of a certain model of dollsproduce 70% and 30% of a certain model of dollsrespectively. It is found that 5% of the dolls produced byrespectively. It is found that 5% of the dolls produced byX and 15% of the dolls produced by Y defective. If a dollX and 15% of the dolls produced by Y defective. If a dollis selected at random, find the probability that the selected is selected at random, find the probability that the selected doll isdoll is(a) produced by X and is not defective,(a) produced by X and is not defective,(b) defective. (b) defective.

G. ExerciseG. Exercise

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