PROBABILITY

Probability

� The likelihood or chance of an event occurring

� If an event is IMPOSSIBLE its probability is ZERO

� If an event is CERTAIN its probability is ONE

� So all probabilities lie between 0 and 1

� Probabilities can be represented as a fraction, decimal of percentages

Probabilty

0 0.5 1

Impossibe Unlikely Equally Likely Likely Certain

Experimental Probability

� Relative Frequency is an estimate of probability

������������ �� =���� ������� �

��������� ��

� Approaches theoretic probability as the number of trials increases

Example

Toss a coin 20 times an observe the relative frequency of getting tails.

Theoretical Probability

� Key Terms:

Each EXPERIMENT has a given number of specific OUTCOMES which

together make up the SAMPLE SPACE(S). The probability of an EVENT (A)

occurring must be such that A is subset of S

� Experiment throwing coin die

� # possible Outcomes, n(S) 2 6

� Sample Space, S H,T 1,2,3,4,5,6

� Event A (A subset S) getting H getting even #

Theoretical Probability

� Probability

The probability of an event A occurring is calculated as:

� � =��������( )

"�#$%��������&�''(�%�)�#*���'=

+( )

+(,)

Examples

1. A fair die is rolled find the probability of getting:a) a “6”

b) a factor of 6

c) a factor of 60

d) a number less than 6

e) a number greater than 6

2. One letter is selected from “excellent”. Find the probability that it is:a) an “e”

b) a consonant

3. One card is selected from a deck of cards find the probability of selecting:a) a Queen

b) a red card

c) a red queen

A B

1

6 4

6=2

36

6= 1

5

6

0

6= 0

3

9=1

3

6

9=2

3

4

52=1

13

26

53=1

2 2

52=1

26

Theoretical Probability

� Conditional Probability

Conditional Probability of A given B is the probability that A occurs given that event B has occurred. This basically changes the sample space to B

� �|6 =+( $+78)

+(9)

Examples

1. A fair die is rolled find the probability of getting:

a) a “6” given that it is an even number

b) a factor of 6, given that it is a factor of 8

2. One letter is selected from “excellent”. Find the probability that it is:

a) a “l” given it is a consonant

b) an “e”, given the letter is in excel

3. One card is selected from a deck of cards find the probability of selecting:

a) a Queen , given it is a face card

b) a red card given it is a queen

c) a queen, given it is red card

A B

1

3

1,2}��<(1,2,4} 2

3

2

6=1

3

{e,e,e} from {e,x,c,e,l,l,e}=

>

4

12=1

32

4=1

2

4

26=2

13

Theoretical Probability

� Expectation

The expectation of an event A is the number of times the event A is expected to

occur within n number of trials,

�(�) = × �(�)

Examples

1. A coin is tossed 30 times. How many time would you expect to get tails?

2. The probability that Mr Bennett wears a blue shirt on a given day is 15%. Find the expected number of days in September that he will wear a blue shirt?

@

A× 30 = 15��<�B

15%× 30 = 4.5 ≈ 5F��B

Sample Space

Sample Space can be represented as:

� List

� Grid/Table

� Two-Way Table

� Venn Diagram

� Tree Diagram

Sample Space

1) LIST:

Bag A: 1 Black , 1 white . Bag B: 1 Black, 1 Red

One marble is selected from each bag.

a) Represent the sample space as a LIST

b) Hence state the probability of choosing the same colours

ANSWER:

G = 66, 6�,H6,H�

� B�<������B =1

4

Sample Space

2) i)GRID:

Two fair dice are rolled and the numbers noted

a) Represent the sample space on a GRID

b) Hence state the probability of choosing the same numbers

ANSWER:

G =

P B�<�#B =K

=K=

@

K

1 2 3 4 5 6

1

2

3

4

5

6

Dice #1

Dice #2

Sample Space

2) ii)TABLE:

Two fair dice are rolled and the sum of the scores is recorded

a) Represent the sample space in a TABLE

b) Hence state the probability of getting an even sum

ANSWER:

G =

P ��� B�< =18

36=1

2

Dice 2\Dice 1 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

Sample Space

3) TWO- WAY TABLE:

A survey of Grade 10 students at a small school returned the following

results:

A student is selected at random, find the probability that:

a) it is a girl

b) the student is not good at math

c) it is a boy who is good at Math

d) it is a girl, given the student is good at Math

e) the student is good at Math, given that it is a girl

Category Boys Girls

Good at Math 17 19

Not good at Math 8 12

P M�� =31

56

25 31 56

36

20

P N��O��F@Q��R =20

56=5

14

P 6��, O��F@Q��R =17

56

P M��|O��F@Q��R =19

36

P O��F@Q��R|M�� =19

31

Sample Space

4) VENN DIAGRAM:

The Venn diagram below shows sports played by students in a class:

A student is selected at random, find the probability that the student:

a) plays basket ball

b) plays basket ball and tennis

c) Plays basketball given that the student plays tennis

P 6�BT��U��� =17

27

P 6�BT��U���&�� �B =4

27

P 6�BT��U���|�� �B =4

11

Sample Space

5) TREE DIAGRAM:

Note: tree diagrams show outcomes and probabilities. The outcome is written at the end of each branch and the probability is written on each branch.

Represent the following in tree diagrams:

a) Two coins are tossed

b) One marble is randomly selected from Bag A with 2 Black & 3 White marbles , then another is selected from Bag B with 5 Black & 2 Red marbles.

c) The state allows each person to try for their pilot license a maximum of 3

times. The first time Mary goes the probability she passes is 45%, if she goes

a second time the probability increases to 53% and on the third chance it

increase to 58%.

Sample Space

5) TREE DIAGRAM:

a) Answer:

Sample Space

5) TREE DIAGRAM:

b) Answer:

Sample Space

5) TREE DIAGRAM:

c) Answer:

Types of Events

� EXHAUSTIVE EVENTS: a set of event are said to be Exhaustive if together

they represent the Sample Space. i.e A,B,C,D are exhaustive if:

P(A)+P(B)+P(C)+P(D) = 1

Eg Fair Dice: P(1)+P(2)+P(3)+P(4)+P(5)+P(6)=

Types of Events

� COMPLEMENTARY EVENTS: two events are said to be complementary if

one of them MUST occur. A’ , read as “A complement” is the event when A

does not occur. A and A’ () are such that: P(A) + P(A’) = 1

� State the complementary event for each of the following

� Eg Find the probability of not getting a 4 when a die is tossed

P(4’) =

� Eg. Find the probability that a card selected at random form a deck of cards is not a queen.

P(Q’)=

A’A

EVENT A A’ (COMPLEMENTARY EVENT)

Getting a 6 on a die

Getting at least a 2 on a die

Getting the same result when a coin is tossed twice

Types of Events

COMPOUND EVENTS:

� EXCLUSIVE EVENTS: a set of event are said to be Exclusive (two events would be

“Mutually Excusive”) if they cannot occur together. i.e they are disjoint sets

� INDEPENDENT EVENTS: a set of event are said to be Independent if the occurrence of

one DOES NOT affect the other.

� DEPENDENT EVENTS: a set of event are said to be dependent if the occurrence of one

DOES affect the other.

A

B

Types of Events

EXCLUSIVE/ INDEPENDENT / DEPENDENT EVENTS

� Which of the following pairs are mutually exclusive events?

Event A Event B

Getting an A* in IGCSE Math Exam Getting an E in IGCSE Math Exam

Leslie getting to school late Leslie getting to school on time

Abi waking up late Abi getting to school on time

Getting a Head on toss 1 of a coin Getting a Tail on toss 1 of a coin

Getting a Head on toss 1 of a coin Getting a Tail on toss 2 of a coin

� Which of the following pairs are dependent/independent events?

Event A Event B

Getting a Head on toss 1 of a coin Getting a Tail on toss 2 of a coin

Alvin studying for his exams Alvin doing well in his exams

Racquel getting an A* in Math Racquel getting an A* in Art

Abi waking up late Abi getting to school on time

Taking Additional Math Taking Higher Level Math

Probabilities of Compound Events

When combining events, one event may or may not have an effect on the other, which

may in turn affect related probabilities

A B

Type of

ProbabilityMeaning Diagram Calculation

AND

X Y ∩ 9

Probability that event A AND event B will occur together.

Generally, AND = multiplication

X Y[\]^_`\9 = X Y × X Y|9

Note:Note:Note:Note:For Exclusive EventsFor Exclusive EventsFor Exclusive EventsFor Exclusive Events:

since they cannot occur together then,

X Y ∩ 9 = a

For IndependentFor IndependentFor IndependentFor Independent: Events:Events:Events:Events:since A is not affected by the occurrence of B

X Y ∩ 9 = X Y × X 9

OR

X Y ∪ 9

Probability that either event A OR event B (or both) will occur.

Generally, OR = addition

X Y ∪ 9 = X Y + X 9 − X Y ∩ 9

Note:Note:Note:Note:For ExclusiveFor ExclusiveFor ExclusiveFor Exclusive Events:Events:Events:Events:since such events are disjoint sets,

X Yef9 = X Y + X 9

A B

A B

Examples – Using “Complementary” Probability

1. The table below show grades of students is a Math Quiz

Find the probability that a student selected at random scored at least 2 on the quiz

(i)By Theoretical Probability (ii) By Complementary

Grade 1 2 3 4 5

Frequency 5 7 10 16 12

Examples – Using “OR” Probability

1. A fair die is rolled, find the probability of getting a 3 or a 5.

(i)By Sample Space (ii) By OR rule

Examples – Using “AND” Probability

1. A fair die is rolled twice find the probability of getting a 5 and a 5.

(i)By Sample Space (ii) By AND rule

Examples – Using “OR” /“AND” Probability

1. A fair die is rolled twice find the probability of getting a 3 and a 5.

(i)By Sample Space (ii) By AND/OR rule

Mixed Examples

1. From a pack of playing cards, 1 card is selected. Find the probability of selecting:

a) A queen or a king

b) Heart or diamond

c) A queen or a heart

d) A queen given that at face card was selected

e) A card that has a value of at least 3 (if face cards have a value of 10 and Ace has a value of 1)

4

52+4

52=8

52

13

52+13

52=1

2

P(Q)+P(H)-P(Q&H)=g

hA+

AK

hA−

@

hA=

A@

hA

4

12=1

3

1 −8

52==

4

52

Mixed Examples

2. From a pack of playing cards, 1 card is selected noted and replaced, then a 2nd card is selected and noted. Find the probability of selecting:

a) A queen and then a king

b) A queen and a king

c) Two cards of same number

d) Two different cards

4

52×4

52=

1

169

P(Q&K) or P(K&Q)=@

@Ki× 2

P(A&A) or P(2&2) or ….PK&K)

=@

@Ki× 13 =

@

@=

1-P(same) =1 −@

@==

@A

@=

Mixed Examples

3. From a pack of playing cards, 1 card is selected noted , it is NOT replaced, then a 2nd card is selected and noted. Find the probability of selecting:

a) A queen and then a king

b) A queen and a king

c) Two cards of same number

d) Two cards with different numbers

Probabilities of Repeated Events

1) A coin is tossed 3 times find the probability of getting: a) tail exactly once

a) a tail AT LEAST once

2) A die is tossed until a 6 appears. Find the probability of getting a 6:

a) on the 2nd tossb) on the 3rd tossc) on the nth toss

1. A die is tossed twice. Draw a tree diagram and find the probability of getting and even number and an odd number.

Tree Diagrams

1.

Tree Diagrams

Tree Diagrams

2.

i) Find the probability that:

a) he is on time for school

b) he is on time everyday in a 5 day week

c) he is on time once in a 5 day week

ii)If there are 60 days this term, how many days would you expect Jack to be late this term?

Tree Diagrams

2.

i) a) b) c)

ii)

Tree Diagrams

3.

Tree Diagrams

3a).

Tree Diagrams

3b)