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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
0.5
0
Impossibe
Unlikely
Equally Likely
1
Likely
Certain
Experimental Probability

Relative Frequency is an estimate of probability
𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝐸𝑣𝑒𝑛𝑡
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 =
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.
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:
𝑃 𝐴 =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 (𝐴)
𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑃𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑂𝑢𝑡𝑐𝑜𝑚𝑒𝑠
=
𝑛(𝐴)
𝑛(𝑆)
Examples
1.
A fair die is rolled find the probability of getting:
a)
b)
c)
d)
e)
2.
5
6
0
=0
6
One letter is selected from “excellent”. Find the probability that it is:
a)
b)
3.
1
a “6” 6
4 2
=
a factor of 6 6 3 6
=1
6
a factor of 60
a number less than 6
a number greater than 6
3 1
an “e”
=
9 3
a consonant
6 2
=
9 3
One card is selected from a deck of cards find the probability of selecting:
a)
b)
c)
a Queen 524 = 131
26 1
a red card
=
53 2
a red queen
2
1
=
52 26
A
B
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)
A
Examples
1.
A fair die is rolled find the probability of getting:
a)
b)
2.
3.
1
3
a “6” given that it is an even number
a factor of 6, given that it is a factor of 8
1,2} 𝑓𝑟𝑜𝑚 (1,2,4}
2
3
One letter is selected from “excellent”. Find the probability that it is:
2 1
=
6 3
a)
a “l” given it is a consonant
b)
an “e”, given the letter is in excel
{e,e,e} from {e,x,c,e,l,l,e}
3
7
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
2 1
=
4 2
4
2
=
26 13
4
1
=
12 3
B
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 × 𝑃(𝐴)
Examples
1.
A coin is tossed 30 times. How many time would you expect to get tails?
1
×
2
2.
30 = 15 𝑡𝑖𝑚𝑒𝑠
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 ≈ 5 𝑑𝑎𝑦𝑠
Sample Space
Sample Space can be represented as:
 List
 Grid/Table
 Two-Way Table
 Venn Diagram
 Tree Diagram
Sample Space
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:
1)
𝑆 = 𝐵𝐵, 𝐵𝑅, 𝑊𝐵, 𝑊𝑅
1
𝑃 𝑠𝑎𝑚𝑒 𝑐𝑜𝑙𝑜𝑢𝑟𝑠 =
4
Sample Space
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:
𝑆=
2)
6
Dice #2
5
4
3
P 𝑠𝑎𝑚𝑒 #𝑠 =
2
1
Dice #1
1
2
3
4
5
6
6
36
=
1
6
Sample Space
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
2)
ANSWER:
𝑆=
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
18 1
P 𝑒𝑣𝑒𝑛 𝑠𝑢𝑚 =
=
36 2
Sample Space
TWO- WAY TABLE:
A survey of Grade 10 students at a small school returned the following
results:
Category
Boys
Girls
3)
Good at Math
17
19
Not good at Math
8
12
25
31
36
20
56
A student is selected at random, find the probability that:
31
P 𝐺𝑖𝑟𝑙 =
a)
it is a girl
56
20
5
P 𝑁𝑜𝑡 𝑔𝑜𝑜𝑑 @𝑀𝑎𝑡ℎ =
=
b)
the student is not good at math
56 14
17
c)
it is a boy who is good at Math
P 𝐵𝑜𝑦, 𝑔𝑜𝑜𝑑 @𝑀𝑎𝑡ℎ =
56
19
P
𝐺𝑖𝑟𝑙|
𝑔𝑜𝑜𝑑
@𝑀𝑎𝑡ℎ
=
d)
it is a girl, given the student is good at Math
36
e)
the student is good at Math, given that it is a girl P 𝑔𝑜𝑜𝑑 @𝑀𝑎𝑡ℎ|𝐺𝑖𝑟𝑙 = 19
31
Sample Space
VENN DIAGRAM:
The Venn diagram below shows sports played by students in a class:
4)
A student is selected at random, find the probability that the student:
17
a)
plays basket ball
P 𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙 =
27
4
b)
plays basket ball and tennis
P 𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙&𝑇𝑒𝑛𝑛𝑖𝑠 =
27
c)
Plays basketball given that the student plays tennis
P 𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙|𝑇𝑒𝑛𝑛𝑖𝑠 =
4
11
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:
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%.
5)
Sample Space
5)
a)
TREE DIAGRAM:
Answer:
Sample Space
5)
b)
TREE DIAGRAM:
Answer:
Sample Space
5)
c)
TREE DIAGRAM:
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
A

State the complementary event for each of the following
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

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’
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
A
B

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.
Types of Events
EXCLUSIVE/ INDEPENDENT / DEPENDENT EVENTS

Which of the following pairs are mutually exclusive events?
Event A
Getting an A* in IGCSE Math Exam
Leslie getting to school late
Abi waking up late
Getting a Head on toss 1 of a coin
Getting a Head on toss 1 of a coin

Event B
Getting an E in IGCSE Math Exam
Leslie getting to school on time
Abi getting to school on time
Getting a Tail 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
Getting a Head on toss 1 of a coin
Alvin studying for his exams
Racquel getting an A* in Math
Abi waking up late
Taking Additional Math
Event B
Getting a Tail on toss 2 of a coin
Alvin doing well in his exams
Racquel getting an A* in Art
Abi getting to school on time
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
Type of
Probability
Meaning
AND
𝑷 𝑨 ∩𝑩
Probability that event A AND event B
will occur together.
Generally,
AND = multiplication
Diagram
Calculation
𝑷 𝑨 𝒂𝒏𝒅 𝒕𝒉𝒆𝒏 𝑩 = 𝑷 𝑨 × 𝑷 𝑨|𝑩
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
B
𝑷 𝑨 ∪ 𝑩 = 𝑷A
𝑨 +𝑷 𝑩 −𝑷 𝑨∩𝑩
Probability that either event A OR
event B (or both) will occur.
Generally,
OR = addition
A
B
Note:
For Exclusive Events:
since such events are disjoint sets,
𝑷 𝑨 𝑶𝑹 𝑩 = 𝑷 𝑨 + 𝑷 𝑩
Examples – Using “Complementary” Probability
1.
The table below show grades of students is a Math
Quiz
Grade
1
2
3
4
5
Frequency
5
7
10
16
12
Find the probability that a student selected at random
scored at least 2 on the quiz
(i)By Theoretical Probability (ii) By Complementary
Examples – Using “OR” Probability
A fair die is rolled, find the probability of getting
a 3 or a 5.
(i)By Sample Space
(ii) By OR rule
1.
Examples – Using “AND” Probability
A fair die is rolled twice find the probability of
getting a 5 and a 5.
(i)By Sample Space
(ii) By AND rule
1.
Examples – Using “OR” /“AND” Probability
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
1.
Mixed Examples
1.
a)
b)
c)
d)
e)
From a pack of playing cards, 1 card is selected.
Find the probability of selecting:
4
4
8
+
=
A queen or a king 52 52 52
Heart or diamond 13 + 13 = 1
52 52 2
1
21
A queen or a heart P(Q)+P(H)-P(Q&H)=524 + 26
−
=
52
52
52
A queen given that at face card was selected 4 = 1
12 3
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)
8
4
1−
==
52
52
Mixed Examples
2.
a)
b)
c)
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:
4
4
1
×
=
A queen and then a king 52 52 169
1
P(Q&K) or P(K&Q)= × 2
169
A queen and a king
1
×
Two cards of same number P(A&A) or P(2&2) or ….PK&K) =169
1
13 = 13
d)
Two different cards
1
12
1-P(same) =1 − 13 = 13
Mixed Examples
3.
a)
b)
c)
d)
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 queen and then a king
A queen and a king
Two cards of same number
Two cards with different numbers
Probabilities of Repeated Events
A coin is tossed 3 times find the probability of getting:
1)
a)
tail exactly once
a)
a tail AT LEAST once
A die is tossed until a 6 appears. Find the probability
of getting a 6:
2)
a)
b)
c)
on the 2nd toss
on the 3rd toss
on the nth toss
Tree Diagrams
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
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)
ii)
b)
c)
Tree Diagrams
3.
Tree Diagrams
3a).
Tree Diagrams
3b)