Note 1 probability

1.Introduction to Probability
Probabilities are associated with experiments where the outcome is not known in
advance or cannot be predicted. For example, if you toss a coin, will you obtain a head
or tail? If you roll a die will obtain 1, 2, 3, 4, 5 or 6? Probability measures and quantifies
"how likely" an event, related to these types of experiment, will happen. The value of a
probability is a number between 0 and 1 inclusive. An event that cannot occur has a
probability (of happening) equal to 0 and the probability of an event that is certain to
occur has a probability equal to 1.(see probability scale below).

In order to quantify probabilities, we need to define the sample space of an experiment and the
events that may be associated with that experiment.

Sample Space and Events
The sample space is the set of all possible outcomes in an experiment.
Example 1: If a die is rolled, the sample space S is given by
S = {1,2,3,4,5,6}
Example 2: If two coins are tossed, the sample space S is given by
S = {HH,HT,TH,TT} , where H = head and T = tail.
Example 3: If two dice are rolled, the sample space S is given by
S = { (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(6,1),(6,2),(6,3),(6,4),(6,5),(6,6) }

We define an event as some specific outcome of an experiment. An event is a subset of
the sample space.
Example 4: A die is rolled (see example 1 above for the sample space). Let us define
event E as the set of possible outcomes where the number on the face of the die is
even. Event E is given by
E = {2,4,6}
Example 5: Two coins are tossed (see example 2 above for the sample space). Let us
define event E as the set of possible outcomes where the number of head obtained is
equal to two. Event E is given by
E = {(HT),(TH)}
Example 6: Two dice are rolled (see example 3 above for the sample space). Let us
define event E as the set of possible outcomes where the sum of the numbers on the
faces of the two dice is equal to four. Event E is given by
E = {(1,3),(2,2),(3,1)}

How to Calculate Probabilities?
1 - Classical Probability Formula: It is based on the fact that all outcomes are equally
likely.
Total number of outcomes in E
P(E)= ________________________________________________
Total number of outcomes in the sample space

Example 7: A die is rolled, find the probability of getting a 3.
The event of interest is "getting a 3". so E = {3}.
The sample space S is given by S = {1,2,3,4,5,6}.
The number of possible outcomes in E is 1 and the number of possible outcomes in S is
6. Hence the probability of getting a 3 is P("3") = 1 / 6.
Example 8: A die is rolled, find the probability of getting an even number.

The event of interest is "getting an even number". so E = {2,4,6}, the even numbers on a
die.
The sample space S is given by S = {1,2,3,4,5,6}.
The number of possible outcomes in E is 3 and the number of possible outcomes in S is
6. Hence the probability of getting a 3 is P("3") = 3 / 6 = 1 / 2.
2 - Empirical Probability Formula: It uses real data on present situations to determine
how likely outcomes will occur in the future. Let us clarify this using an example
30 people were asked about the colors they like and here are the results:
Color

frequency

red

10

blue

15

green

5

If a person is selected at random from the above group of 30, what is the probability that this
person likes the red color?
Let event E be "likes the red color". Hence
Frequency for red color
P(E)= ________________________________________________
Total frequencies in the above table

= 10 / 30 = 1 / 3

Example 8: The table below shows students distribution per grade in a school.
Grade frequency
1

50

2

30

3

40

4

42

5

38

6

50

If a student is selected at random from this school, what is the probability that this student is in
grade 3?
Let event E be "student from grade 3". Hence
Frequency for grade 3
P(E)= _______________________________________
Total frequencies

= 40 / 250 = 0.16

2. Mutually Exclusive Events - Examples
With Solutions
Two events are mutually exclusive if they cannot occur at the same time.
Using Venn diagram, two events that are mutually exclusive may be represented as
follows:

The two events are such that
E1 ∩ E2 = Φ

The two sets E1 and E2 have no elements in common and their intersection is an empty
set since they cannot occur at the same time.
Using Venn diagram, two events that are not mutually exclusive may be represented
as follows:

E1 ∩ E2 = {c} , the intersection of the two events E1 and E2 is not an empty set
Example 1:
A die is rolled. Let us define event E1 as the set of possible outcomes where the
number on the face of the die is even and event E2 as the set of possible outcomes
where the number on the face of the die is odd. Are event1 E1 and E2 mutually
exclusive?
Solution to Example 1:
•

We first list the elements of E1 and E2.
E1 = {2,4,6}
E2 = {1,3,5}

•

E1 and E2 have no elements in common and therefore are mutually exclusive.

•

Another way to answer the above question is to note is that if you roll a die, it
shows a number that is either even or odd but no number will be even and odd at
the same time. Hence E1 and E2 cannot occur at the same time and are therfore
mutually exclusive.

Example 2:
A die is rolled. Event E1 is the set of possible outcomes where the number on the face
of the die is even and event E2 as the set of possible outcomes where the number on
the face of the die is greater than 3. Are event E1 and E2 mutually exclusive?

•

The subsets E1 and E2 are given by.
E1 = {2,4,6}
E2 = {4,5,6}

•

Subsets E1 and E2 have 2 elements in common. If the die shows 4 or 6, both
events E1 and E2 will have occured at the same time and therefore E1 and E2
are not mutually exclusive.

Example 3:
A card is drawn from a deck of cards. Events E1, E2, E3, E4 and E5 are defined as
follows:
E1: Getting an 8
E2: Getting a king
E3: Getting a face card
E4: Getting an ace
E5: Getting a heart
a) Are events E1 and E2 mutually exclusive?
b) Are events E2 and E3 mutually exclusive?
c) Are events E3 and E4 mutually exclusive?
d) Are events E4 and E5 mutually exclusive?
e) Are events E5 and E1 mutually exclusive?
•

The sample space of the experiment "card is drawn from a deck of cards" is
shown below.

•

a) E1 and E2 are mutually exclusive because there are no cards with an 8 and
a king together.

•

b) E2 and E3 are not mutually exclusive because a king is a face card.

•

c) E3 and E4 are mutually exclusive because an ace is not a face card.

•

d) E4 and E5 are not mutually exclusive because there is one card that has an
ace and a heart.

•

d) E5 and E1 are not mutually exclusive because there is one card that is an 8
of heart.

Example 4: Two dice are rolled. We define events E1, E2, E3 and E4 as follows
E1: Getting a sum equal to 10
E2: Getting a double
E3: Getting a sum less than 4
E4: Getting a sum less to 7
a) Are events E1 and E2 mutually exclusive?
b) Are events E2 and E3 mutually exclusive?
c) Are events E3 and E4 mutually exclusive?

d) Are events E4 and E1 mutually exclusive?
•

The sample space of the experiment "2 dice" is shown below.

•

a) E1 and E2 are not mutually exclusive because outcome (5,5) is a double
and also gives a sum of 10. The two events may occur at the same time.

•

b) E2 and E3 are not mutually exclusive because outcome (1,1) is a double
and gives a sum of 2 and is less than 4. The two events E2 and E3 may occur at
the same time.

•

c) E3 and E4 are not mutually exclusive a sum can be less than 7 and less
than 4 a the same time. Example outcome (1,2).

•

d) E4 and E1 are mutually exclusive because a sum less than 7 cannot be
equal to 10 at the same time. The two events cannot occur at the same time.

2.1 SET IDENTITIES
The following identities can be used if there is a need.

1.
2.
3.
4.
5.
6.
7.

A∪ A = A
A ∪φ = φ
A∪S = S
A∩ A = A
A∪ B = B ∪ A
A∩S = A
A ∩φ = φ

8. A ∩ B = B ∩ A
9. A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C )
10. A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )
11. ( A ∪ B )' = A∩' B '
12. ( A ∩ B )' = A'∪ B '
13. A ∪ B = A ∪ ( A'∩ B )
14. B = ( A ∩ B ) ∪ ( A'∩ B )

Example 1

1. A = {1, 2, 3, 4, 5, 6} and B = {1, 3, 6, 7}, find:
i) A ∩ B
ii) A ∪ B
iii) n( A ∪ B )
2. A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6} and C = {3, 5, 7, 9}, find:
i) A ∩ B
ii) A ∩ C
iii) B ∩ C
iv) A ∩ B ∩ C
v) n ( A ∩ B ∩ C )
3. Based on the Venn diagram below, A and B are two events in the sample
space S. Find:

S

40
B
A 35

a.
b.
c.
d.
e.
f.

P(A)
P(A’)
P(B)
P(B’)
P( A ∩ B)
P( A ∩ B' )

5

20

(g) P ( A'∩ B )
(h) P ( A'∩ B ' )
(i) P ( A ∪ B )
(j) P ( A ∪ B ' )
(k) P ( A'∪ B )
(l) P ( A'∪ B ' )

Solution to example 1.

1.

i) A ∩ B = {1, 3, 6}
ii) A ∪ B = {1, 2, 3, 4, 5, 6, 7}
iii) n ( A ∪ B ) = n(A) + n(B) – n ( A ∩ B ) = 6 + 4 – 3 = 7
*If A, B and C are finite sets, therefore:
n( A ∪ B ∪ C ) = n( A) + n( B ) + n(C ) − n( A ∩ B ) − n( A ∩ C ) − n( B ∩ C ) + n( A ∩ B ∩ C )

2.

i)
ii)
iii)
iv)
v)

A ∩ B = {2, 4, 6}
A ∩ C = {3, 5}
B ∩ C = { φ}
A ∩ B ∩ C = { φ}
( A ∩ B ∩ C)
n( A) + n( B ) + n(C ) − n( A ∩ B ) − n( A ∩ C ) − n( B ∩ C ) + n( A ∩ B ∩ C )
=6+3+4–3–2–0+0
=8

3.

n(S) = 35 + 5 + 20 + 40
n( A) 40 2
=
=
n( S ) 100 5
n( A' ) 60 3
b. n(A’) = 60 and P(A’) =
=
=
n( S ) 100 5
n( B ) 25 1
c. n(B) = 25 and P(B) =
=
=
n( S ) 100 4
n( B ' ) 75 3
d. n(B’) = 75 and P(B’) =
=
=
n( S ) 100 4
n( A ∩ B )
5
1
e. n(A ∩ B) = 5 and P(A ∩ B) =
=
=
n( S )
100 20
n( A ∩ B ' ) 35
7
=
=
f. n(A ∩ B’) = 35 and P(A ∩ B ’) =
n( S )
100 20
P ( A'∩ B ) 20 1
g. n(A’ ∩ B ) = 20 and P(A’ ∩ B) =
=
=
n( S )
100 5
P ( A'∩ B ' ) 40 2
h. n(A’ ∩ B’) = 40 and P(A’ ∩ B’) =
=
=
n( S )
100 5
P ( A ∪ B ) 60 3
i. n(A ∪ B) = 60 and P(A ∪ B) =
=
=
n( S )
100 5
P ( A ∪ B ' ) 80 4
j. n(A ∪ B’) = 80 and P(A ∪ B’) =
=
=
n( S )
100 5
P ( A'∪ B ) 65 13
k. n(A’ ∪ B ) = 65 and P(A’ ∪ B ) =
=
=
n( S )
100 20
n( A'∪ B ' ) 95 19
l. n(A’ ∪ B ' ) = 95 and P(A’ ∪ B’) =
=
=
n( S )
100 20
a. n(A) = 40 and P(A) =

Note 1 probability

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