PROBABILITY

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 1Impossibe Unlikely Equally Likely Likely Certain

Experimental Probability Relative Frequency is an estimate of

probability

Approaches theoretic probability as the number of trials increases

ExampleToss 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:

Examples1. A fair die is rolled find the probability of getting:

a) a “6”b) a factor of 6c) a factor of 60d) a number less than 6e) 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 Queenb) a red cardc) a red queen

A B

16 46=

2366=15606=0

39=

1369=

23

452=

1132653=

12252=

126

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

Examples1. A fair die is rolled find the probability of getting:

a) a “6” given that it is an even numberb) 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 consonantb) 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 cardb) a red card given it is a queenc) a queen, given it is red card

A B

13

¿26=

13

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

412=

132

4=12

426=

213

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,

Examples1. 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?

15%×30=4.5≈5𝑑𝑎𝑦𝑠

Sample SpaceSample Space can be represented as: List Grid/Table Two-Way Table Venn Diagram Tree Diagram

Sample Space1) LIST:Bag A: 1 Black , 1 white . Bag B: 1 Black, 1 RedOne marble is selected from each bag. a) Represent the sample space as a LISTb) Hence state the probability of choosing

the same coloursANSWER:

Sample Space2) i)GRID:Two fair dice are rolled and the numbers noteda) Represent the sample space on a GRIDb) Hence state the probability of choosing the

same numbersANSWER:

1 2 3 4 5 6

1

2

3

4

5

6

Dice #1

Dice #2

Sample Space2) ii)TABLE:Two fair dice are rolled and the sum of the scores is recordeda) Represent the sample space in a TABLEb) Hence state the probability of getting

an even sumANSWER:

Dice 2\Dice 1 1 2 3 4 5 6

1 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12

Sample Space3) 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 mathc) it is a boy who is good at Mathd) it is a girl, given the student is good at Mathe) the student is good at Math, given that it is a girl

Category Boys GirlsGood at

Math 17 19

Not good at Math 8 12P (𝐺𝑖𝑟𝑙 )=3156

25 31 56

36 20

P (𝑁𝑜𝑡 𝑔𝑜𝑜𝑑@ h𝑀𝑎𝑡 )=2056=514

P (𝐵𝑜𝑦 ,𝑔𝑜𝑜𝑑@ h𝑀𝑎𝑡 )=1756P (𝐺𝑖𝑟𝑙∨𝑔𝑜𝑜𝑑@ h𝑀𝑎𝑡 )=1936P (𝑔𝑜𝑜𝑑@ h𝑀𝑎𝑡 ∨𝐺𝑖𝑟𝑙 )=1931

Sample Space4) 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 ballb) plays basket ball and tennisc) Plays basketball given that the student plays tennis

P (𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙 )=1727P (𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙∧𝑇𝑒𝑛𝑛𝑖𝑠 )= 4

27P (𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙∨𝑇𝑒𝑛𝑛𝑖𝑠 )= 4

11

Sample Space5) 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 tossedb) 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 Space5) TREE DIAGRAM:a) Answer:

Sample Space5) TREE DIAGRAM:b) Answer:

Sample Space5) 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 dieGetting at least a 2 on a dieGetting the same result when a coin is tossed twice

Types of EventsCOMPOUND 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.

AB

Types of EventsEXCLUSIVE/ 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 ExamLeslie getting to school late Leslie getting to school on timeAbi waking up late Abi getting to school on timeGetting a Head on toss 1 of a coin Getting a Tail on toss 1 of a coinGetting 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 coinAlvin studying for his exams Alvin doing well in his examsRacquel getting an A* in Math Racquel getting an A* in ArtAbi waking up late Abi getting to school on timeTaking Additional Math Taking Higher Level Math

Probabilities of Compound EventsWhen 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 Probability

Meaning Diagram Calculation

AND Probability that event A AND event B will occur together.

Generally, AND = multiplication

Note:For Exclusive Events: since they cannot occur together then,

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

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

Generally, OR = addition

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

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 5Frequency

5 7 10 16 12

Examples – Using “OR” Probability1. 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” Probability1. 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 Examples1. From a pack of playing cards, 1 card is

selected. Find the probability of selecting:

a) A queen or a kingb) Heart or diamondc) A queen or a heartd) A queen given that at face card was

selectede) 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)

452+

452=

852

1352+

1352=

12

P(Q)+P(H)-P(Q&H)=412=

13

1− 852=¿452

Mixed Examples2. 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 kingb) A queen and a kingc) Two cards of same number

d) Two different cards

452×

452=

1169

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

P(A&A) or P(2&2) or ….PK&K) =

1-P(same) =

Mixed Examples3. 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 kingb) A queen and a kingc) Two cards of same numberd) Two cards with different numbers

Probabilities of Repeated Events1) A coin is tossed 3 times find the probability of getting:

a) tail exactly once

b) 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 Diagrams2.

i) Find the probability that:a) he is on time for schoolb) he is on time everyday in a 5 day weekc) 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 Diagrams2.

i) a) b) c)ii)

Tree Diagrams3.

Tree Diagrams3a).

Tree Diagrams3b)