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

of 41/41
PROBABILITY
  • date post

    20-Jun-2020
  • Category

    Documents

  • view

    31
  • download

    1

Embed Size (px)

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

    ANSWER:

    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

    ANSWER:

    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

    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��[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:

    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=

    [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)