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

Theoretical Probability

Key Terms: 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 #

Probability The probability of an event A occurring is calculated as: P(A)= #A/#Outcomes Examples One letter selected from excellent consonant dice   Expectation The expectation of an event A is the number of times the event A is

expected to occur within n number of trials, E(A)=n x P(A)   coin , 30 tosses expectd # tails rainfall = 20% , expected # days in sept?

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 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 Space

2) 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 Space

2) 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 sum

ANSWER:

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 mathc) It is a boy who is good at Math

Category Boys Girls

Good at Math 17 19

Not good at Math 8 12

P (𝐺𝑖𝑟𝑙 )=3156

25 31 56

25

20

P (𝑁𝑜𝑡 𝑔𝑜𝑜𝑑@ h𝑀𝑎𝑡 )=2056

=514

P (𝐵𝑜𝑦 ,𝑔𝑜𝑜𝑑@ h𝑀𝑎𝑡 )=1756

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 ballb) plays basket ball and tennis

P (𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙 )=1727

P (𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙∧𝑇𝑒𝑛𝑛𝑖𝑠 )= 456

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 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 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 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.

A

B

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 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 Probability

Meaning Diagram Calculation

CONDITIONAL

A given B

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

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

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 “Conditional” Probability

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

A student selected at random, Given that the student scored more than 3,find the probability that he/she scored 5

Grade 1 2 3 4 5

Frequency

5 7 10 16 12

Examples- Conditional Probability

a.b.

BB’

M F

155 145 300

140

160

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” 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 Examples

1. 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)

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 kingb) A queen and a kingc) Heart or diamondd) Two cards of same numbere) Two different cards

All the hungry-bellies began begging for free food (especially Leslie and Samantha). So we did not get to finish these questions

Please write out the remaining examples and leave space for us to discuss tomorrow

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 kingb) A queen and a kingc) Heart or diamondd) Two cards of same numbere) Two cards with different numbers

Using Tree Diagrams