1. PROBABILITY

2. 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
Impossibe Unlikely Equally Likely Likely Certain

3. Experimental Probability
Relative Frequency is an estimate of probability
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦= 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝐸𝑣𝑒𝑛𝑡 𝑇𝑜𝑡𝑎𝑙 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
Approaches theoretic probability as the number of trials increases
Example
Toss a coin 20 times an observe the relative frequency of getting tails.

4. 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 #

5. Theoretical Probability
The probability of an event A occurring is calculated as:
𝑃 𝐴 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 (𝐴) 𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑃𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑂𝑢𝑡𝑐𝑜𝑚𝑒𝑠 = 𝑛(𝐴) 𝑛(𝑆)
Examples
A fair die is rolled find the probability of getting:
a “6”
a factor of 6
a factor of 60
a number less than 6
a number greater than 6
One letter is selected from “excellent”. Find the probability that it is:
an “e”
a consonant
One card is selected from a deck of cards find the probability of selecting:
a Queen
a red card
a red queen A B
1 6
4 6 = 2 3
6 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

6. 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
𝑃 𝐴|𝐵 = 𝑛(𝐴𝑎𝑛𝑑 𝐵) 𝑛(𝑩)
Examples
A fair die is rolled find the probability of getting:
a “6” given that it is an even number
a factor of 6, given that it is a factor of 8
One letter is selected from “excellent”. Find the probability that it is:
a “l” given it is a consonant
an “e”, given the letter is in excel
One card is selected from a deck of cards find the probability of selecting:
a Queen , given it is a face card
a red card given it is a queen
a queen, given it is red card A B
1 3
1,2} 𝑓𝑟𝑜𝑚 (1,2,4}
2 6 = 1 3
{e,e,e} from {e,x,c,e,l,l,e}
4 12 = 1 3
2 4 = 1 2
4 26 = 2 13

7. 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,
𝐸(𝐴)=𝑛 𝑥 𝑃(𝐴)
Examples
A coin is tossed 30 times. How many time would you expect to get tails?
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?
1 2 ×30 =15 𝑡𝑖𝑚𝑒𝑠
15%×30=4.5 ≈5 𝑑𝑎𝑦𝑠

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

9. Sample Space LIST: Bag A: 1 Black , 1 white . Bag B: 1 Black, 1 Red
One marble is selected from each bag.
Represent the sample space as a LIST
Hence state the probability of choosing the same colours
ANSWER:
𝑆= 𝐵𝐵,𝐵𝑅,𝑊𝐵,𝑊𝑅
𝑃 𝑠𝑎𝑚𝑒 𝑐𝑜𝑙𝑜𝑢𝑟𝑠 = 1 4

10. Sample Space i)GRID: Two fair dice are rolled and the numbers noted
Represent the sample space on a GRID
Hence state the probability of choosing the same numbers
ANSWER: 𝑆=
P 𝑠𝑎𝑚𝑒 #𝑠 = 6 36 = 1 6

11. Sample Space ANSWER: ii)TABLE:
Two fair dice are rolled and the sum of the scores is recorded
Represent the sample space in a TABLE
Hence state the probability of getting an even sum
ANSWER: 𝑆=
P 𝑒𝑣𝑒𝑛 𝑠𝑢𝑚 = = 1 2
Dice 2\Dice 1 1 2 3 4 5 6 7 8 9 10 11 12

12. Sample Space 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:
it is a girl
the student is not good at math
it is a boy who is good at Math
it is a girl, given the student is good at Math
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 𝐺𝑖𝑟𝑙 = 31 56
P 𝑁𝑜𝑡 = = 5 14
P 𝐵𝑜𝑦, = 17 56
P 𝐺𝑖𝑟𝑙| = 19 36
P = 19 31

13. Sample Space 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:
plays basket ball
plays basket ball and tennis
Plays basketball given that the student plays tennis
P 𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙 = 17 27
P 𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙&𝑇𝑒𝑛𝑛𝑖𝑠 = 4 56
P 𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙|𝑇𝑒𝑛𝑛𝑖𝑠 = 4 17

14. Sample Space 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:
Two coins are tossed
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.
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%.

15. Sample Space
TREE DIAGRAM:
Answer:

16. Sample Space
TREE DIAGRAM:
Answer:

17. Sample Space
TREE DIAGRAM:
Answer:

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

19. 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

20. 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

21. 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?
Alvin studying for his exams Alvin doing well in his exams
Racquel getting an A* in Math Racquel getting an A* in Art
Taking Additional Math Taking Higher Level Math

22. 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
Type of Probability
Meaning
Diagram
Calculation
AND
𝑷 𝑨 ∩𝑩
Probability that event A AND event B will occur together.
Generally,
AND = multiplication
𝑷 𝑨 𝒂𝒏𝒅 𝒕𝒉𝒆𝒏 𝑩 =𝑷 𝑨 ×𝑷 𝑨|𝑩
Note:
For Exclusive Events:
since they cannot occur together then,
𝑷 𝑨 ∩ 𝑩 =𝟎
For Independent: Events:
since A is not affected by the occurrence of B
𝑷 𝑨 ∩ 𝑩 =𝑷 𝑨 ×𝑷 𝑩 OR
𝑷 𝑨 ∪𝑩
Probability that either event A OR event B (or both) will occur.
OR = addition
𝑷 𝑨 ∪𝑩 =𝑷 𝑨 +𝑷 𝑩 −𝑷 𝑨∩𝑩
since such events are disjoint sets,
𝑷 𝑨 𝑶𝑹 𝑩 =𝑷 𝑨 +𝑷 𝑩 A B A B A B