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PROBABILITY
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### Transcript of PROBABILITY - Edwards E Z Mathedwardsezmath.weebly.com/uploads/1/3/5/1/13518726/... · Theoretical...

• 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��

• 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

G = 66, 6�,H6,H�

� B�

• 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

G =

P B�

• 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

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��[email protected]��R =20

56=5

14

P 6��, O��[email protected]��R =17

56

P M��|O��[email protected]��R =19

36

P O��[email protected]��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:

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:

• Sample Space

5) TREE DIAGRAM:

• Sample Space

5) TREE DIAGRAM:

• 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.

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=

[email protected]

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)[email protected]

@Ki× 2

P(A&A) or P(2&2) or ….PK&K)

[email protected]

@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)