Page 1
Probability and Statistics
Basic Terms
Random Experiment
Consider an action which is repeated under essentially identical conditions. If it results in
any one of the several possible outcomes, but it is not possible to predict which outcome
will appear. Such an action is called as a Random Experiment. One performance of such
an experiment is called as a Trial.
Sample Space
The set of all possible outcomes of a random experiment is called as the sample space.
All the elements of the sample space together are called as ‘exhaustive cases’. The
number of elements of the sample space i.e. the number of exhaustive cases is denoted
by n(S) or N or n.
Event
Any subset of the sample space is called as an ‘Event’ and is denoted by some capital
letter like A, B, C or A
1
, A
2
, A
3
,.. or B
1
, B
2
, ... etc.
Favourable cases
The cases which ensure the happening of an event A, are called as the cases favourable
to the event A. The number of cases favourable to event A is denoted by n(A) or N
A
or n
A
.
Mutually Exclusive Events or Disjoint Events
Two events A and B are said to be mutually exclusive or disjoint if A ? B = ?
i.e. if there is no element common to A & B.
Equally Likely Cases
Cases are said to be equally likely if they all have the same chance of occurrence i.e. no
case is preferred to any other case.
Permutation
A permutation is an arrangement of all or part of a set of objects. The number of
permutations of n distinct objects taken r at a time is
n
P
r
=
??
n!
nr! ?
Page 2
Probability and Statistics
Basic Terms
Random Experiment
Consider an action which is repeated under essentially identical conditions. If it results in
any one of the several possible outcomes, but it is not possible to predict which outcome
will appear. Such an action is called as a Random Experiment. One performance of such
an experiment is called as a Trial.
Sample Space
The set of all possible outcomes of a random experiment is called as the sample space.
All the elements of the sample space together are called as ‘exhaustive cases’. The
number of elements of the sample space i.e. the number of exhaustive cases is denoted
by n(S) or N or n.
Event
Any subset of the sample space is called as an ‘Event’ and is denoted by some capital
letter like A, B, C or A
1
, A
2
, A
3
,.. or B
1
, B
2
, ... etc.
Favourable cases
The cases which ensure the happening of an event A, are called as the cases favourable
to the event A. The number of cases favourable to event A is denoted by n(A) or N
A
or n
A
.
Mutually Exclusive Events or Disjoint Events
Two events A and B are said to be mutually exclusive or disjoint if A ? B = ?
i.e. if there is no element common to A & B.
Equally Likely Cases
Cases are said to be equally likely if they all have the same chance of occurrence i.e. no
case is preferred to any other case.
Permutation
A permutation is an arrangement of all or part of a set of objects. The number of
permutations of n distinct objects taken r at a time is
n
P
r
=
??
n!
nr! ?
Notes on Probability and Statistics
Note: The number of permutations of n distinct objects is n! i.e.,
n
P
n
= n!
The number of permutations of n distinct objects arranged in a circle
is (n ? 1)!
The number of distinct permutations of n things of which n
1
are of one
kind, n
2
of a second kind … n
k
of a k
th
kind is
12 k
n!
n ! n ! ......n !
Combination
A combination is selection of all or part of a set of objects. The number of combinations of
n distinct objects taken r at a time is
??
n
r
n!
C=
r! n r ! ?
Note: In a permutation, the order of arrangement of the objects is important.
Thus abc is a different permutation from bca.
In a combination, the order in which objects are selected does not matter.
Thus abc and bca are the same combination.
Definition of Probability
Consider a random experiment which results in a sample space containing n(S) cases
which are exhaustive, mutually exclusive and equally likely. Suppose, out of n(S) cases,
n(A) cases are favourable to an event A. Then the probability of event A is denoted by
P(A) and is defined as follows.
P(A) =
n(A)
n(S)
=
number of cases favourable to event A
number of cases in the sample space S
Complement of an event
The complement of an event A is denoted by A and it contains all the elements of the
sample space which do not belong to A.
For example: Random experiment: an unbiased die is rolled.
S = {1, 2, 3, 4, 5, 6}
(i) Let A: number on the die is a perfect square
? A = {1, 4} ? A = {2, 3, 5, 6}
Page 3
Probability and Statistics
Basic Terms
Random Experiment
Consider an action which is repeated under essentially identical conditions. If it results in
any one of the several possible outcomes, but it is not possible to predict which outcome
will appear. Such an action is called as a Random Experiment. One performance of such
an experiment is called as a Trial.
Sample Space
The set of all possible outcomes of a random experiment is called as the sample space.
All the elements of the sample space together are called as ‘exhaustive cases’. The
number of elements of the sample space i.e. the number of exhaustive cases is denoted
by n(S) or N or n.
Event
Any subset of the sample space is called as an ‘Event’ and is denoted by some capital
letter like A, B, C or A
1
, A
2
, A
3
,.. or B
1
, B
2
, ... etc.
Favourable cases
The cases which ensure the happening of an event A, are called as the cases favourable
to the event A. The number of cases favourable to event A is denoted by n(A) or N
A
or n
A
.
Mutually Exclusive Events or Disjoint Events
Two events A and B are said to be mutually exclusive or disjoint if A ? B = ?
i.e. if there is no element common to A & B.
Equally Likely Cases
Cases are said to be equally likely if they all have the same chance of occurrence i.e. no
case is preferred to any other case.
Permutation
A permutation is an arrangement of all or part of a set of objects. The number of
permutations of n distinct objects taken r at a time is
n
P
r
=
??
n!
nr! ?
Notes on Probability and Statistics
Note: The number of permutations of n distinct objects is n! i.e.,
n
P
n
= n!
The number of permutations of n distinct objects arranged in a circle
is (n ? 1)!
The number of distinct permutations of n things of which n
1
are of one
kind, n
2
of a second kind … n
k
of a k
th
kind is
12 k
n!
n ! n ! ......n !
Combination
A combination is selection of all or part of a set of objects. The number of combinations of
n distinct objects taken r at a time is
??
n
r
n!
C=
r! n r ! ?
Note: In a permutation, the order of arrangement of the objects is important.
Thus abc is a different permutation from bca.
In a combination, the order in which objects are selected does not matter.
Thus abc and bca are the same combination.
Definition of Probability
Consider a random experiment which results in a sample space containing n(S) cases
which are exhaustive, mutually exclusive and equally likely. Suppose, out of n(S) cases,
n(A) cases are favourable to an event A. Then the probability of event A is denoted by
P(A) and is defined as follows.
P(A) =
n(A)
n(S)
=
number of cases favourable to event A
number of cases in the sample space S
Complement of an event
The complement of an event A is denoted by A and it contains all the elements of the
sample space which do not belong to A.
For example: Random experiment: an unbiased die is rolled.
S = {1, 2, 3, 4, 5, 6}
(i) Let A: number on the die is a perfect square
? A = {1, 4} ? A = {2, 3, 5, 6}
(ii) Let B: number on the die is a prime number
? B = {2, 3, 5} ?B = {1, 4, 6}
Note: P(A) + P( A ) = 1 i.e. P(A) = 1 ? P( A )
For any events A and B, ?? ? ? ? ?
?? PA=PAB+PAB
Independent Events
Two events A & B are said to be independent if
P(A ? B) = P(A).P(B)
Note: If A & B are independent then
A & B are independent
A & B are independent
A &B are independent
Theorems of Probability
Addition Theorem
If A and B are any two events then
P(A ?B) = P(A) + P(B) – P(A ?B)
Note: 1. A ? B : either A or B or both i.e. at least one of A & B
AB ? : neither A nor B i.e. none of A & B
A ?B & AB ? are complement to each other
? P(AB ? ) = 1 – P(A ? B)
2. If A & B are mutually exclusive, P(A ? B) = 0
?P(A ? B) = P(A) + P(B)
3. ?? ? ? ? ? ? ? ? ? ?? ?? ? ?
1 2 3 1 231 2
PA A A =P A +PA +PA P A A
? ? ? ? ? ? ??? ? ? ? ?
23 3 1 1 23
PA A P A A +PA A A
Page 4
Probability and Statistics
Basic Terms
Random Experiment
Consider an action which is repeated under essentially identical conditions. If it results in
any one of the several possible outcomes, but it is not possible to predict which outcome
will appear. Such an action is called as a Random Experiment. One performance of such
an experiment is called as a Trial.
Sample Space
The set of all possible outcomes of a random experiment is called as the sample space.
All the elements of the sample space together are called as ‘exhaustive cases’. The
number of elements of the sample space i.e. the number of exhaustive cases is denoted
by n(S) or N or n.
Event
Any subset of the sample space is called as an ‘Event’ and is denoted by some capital
letter like A, B, C or A
1
, A
2
, A
3
,.. or B
1
, B
2
, ... etc.
Favourable cases
The cases which ensure the happening of an event A, are called as the cases favourable
to the event A. The number of cases favourable to event A is denoted by n(A) or N
A
or n
A
.
Mutually Exclusive Events or Disjoint Events
Two events A and B are said to be mutually exclusive or disjoint if A ? B = ?
i.e. if there is no element common to A & B.
Equally Likely Cases
Cases are said to be equally likely if they all have the same chance of occurrence i.e. no
case is preferred to any other case.
Permutation
A permutation is an arrangement of all or part of a set of objects. The number of
permutations of n distinct objects taken r at a time is
n
P
r
=
??
n!
nr! ?
Notes on Probability and Statistics
Note: The number of permutations of n distinct objects is n! i.e.,
n
P
n
= n!
The number of permutations of n distinct objects arranged in a circle
is (n ? 1)!
The number of distinct permutations of n things of which n
1
are of one
kind, n
2
of a second kind … n
k
of a k
th
kind is
12 k
n!
n ! n ! ......n !
Combination
A combination is selection of all or part of a set of objects. The number of combinations of
n distinct objects taken r at a time is
??
n
r
n!
C=
r! n r ! ?
Note: In a permutation, the order of arrangement of the objects is important.
Thus abc is a different permutation from bca.
In a combination, the order in which objects are selected does not matter.
Thus abc and bca are the same combination.
Definition of Probability
Consider a random experiment which results in a sample space containing n(S) cases
which are exhaustive, mutually exclusive and equally likely. Suppose, out of n(S) cases,
n(A) cases are favourable to an event A. Then the probability of event A is denoted by
P(A) and is defined as follows.
P(A) =
n(A)
n(S)
=
number of cases favourable to event A
number of cases in the sample space S
Complement of an event
The complement of an event A is denoted by A and it contains all the elements of the
sample space which do not belong to A.
For example: Random experiment: an unbiased die is rolled.
S = {1, 2, 3, 4, 5, 6}
(i) Let A: number on the die is a perfect square
? A = {1, 4} ? A = {2, 3, 5, 6}
(ii) Let B: number on the die is a prime number
? B = {2, 3, 5} ?B = {1, 4, 6}
Note: P(A) + P( A ) = 1 i.e. P(A) = 1 ? P( A )
For any events A and B, ?? ? ? ? ?
?? PA=PAB+PAB
Independent Events
Two events A & B are said to be independent if
P(A ? B) = P(A).P(B)
Note: If A & B are independent then
A & B are independent
A & B are independent
A &B are independent
Theorems of Probability
Addition Theorem
If A and B are any two events then
P(A ?B) = P(A) + P(B) – P(A ?B)
Note: 1. A ? B : either A or B or both i.e. at least one of A & B
AB ? : neither A nor B i.e. none of A & B
A ?B & AB ? are complement to each other
? P(AB ? ) = 1 – P(A ? B)
2. If A & B are mutually exclusive, P(A ? B) = 0
?P(A ? B) = P(A) + P(B)
3. ?? ? ? ? ? ? ? ? ? ?? ?? ? ?
1 2 3 1 231 2
PA A A =P A +PA +PA P A A
? ? ? ? ? ? ??? ? ? ? ?
23 3 1 1 23
PA A P A A +PA A A
Notes on Probability and Statistics
Multiplication Theorem
If A & B are any two events then
P(A ? B) = P(A).P(B/A) = P(B).P(A/B)
1. Conditional probability of occurrence of event B given that event A has already
occurred.
P(B/A) =
? ?
??
PA B
PA
?
2. Conditional probability of occurrence of event A given that event B has already
occurred
P(A/B) =
? ?
??
PA B
PB
?
Bayes’ Theorem
Suppose that a sample space S is a union of mutually disjoint events B
1
, B
2
, B
3
, ..., B
n
,
suppose A is an event in S, and suppose A and all the B
i
’s have nonzero probabilities. If
k is an integer with 1 = k = n, then
P(B
k
/ A) =
? ? ? ?
? ??? ???? ????
kk
11 2 2 n n
PA/B P B
P A / B P B +P A / B P B +...+P A / B P B
Solved Example 1 :
A single die is tossed. Find the probability
of a 2 or 5 turning up.
Solution :
The sample space is S = {1, 2, 3, 4, 5, 6}
P(1) = P(2) = … = P(6) =
1
6
The event that either 2 or 5 turns up in
indicated by 2 ? 5. Thus
? ? ?? ??
11 1
P2 5 P 2 P 5
66 3
?? ? ? ? ?
Solved Example 2 :
A coin is tossed twice. What is the
probability that at least one head occurs ?
Solution :
The sample space is
S = {HH, HT, TH, TT}
Probability of each outcomes = 1/4
Probability of atleast one head occurring is
P(A) =
111 3
44 4 4
? ??
Page 5
Probability and Statistics
Basic Terms
Random Experiment
Consider an action which is repeated under essentially identical conditions. If it results in
any one of the several possible outcomes, but it is not possible to predict which outcome
will appear. Such an action is called as a Random Experiment. One performance of such
an experiment is called as a Trial.
Sample Space
The set of all possible outcomes of a random experiment is called as the sample space.
All the elements of the sample space together are called as ‘exhaustive cases’. The
number of elements of the sample space i.e. the number of exhaustive cases is denoted
by n(S) or N or n.
Event
Any subset of the sample space is called as an ‘Event’ and is denoted by some capital
letter like A, B, C or A
1
, A
2
, A
3
,.. or B
1
, B
2
, ... etc.
Favourable cases
The cases which ensure the happening of an event A, are called as the cases favourable
to the event A. The number of cases favourable to event A is denoted by n(A) or N
A
or n
A
.
Mutually Exclusive Events or Disjoint Events
Two events A and B are said to be mutually exclusive or disjoint if A ? B = ?
i.e. if there is no element common to A & B.
Equally Likely Cases
Cases are said to be equally likely if they all have the same chance of occurrence i.e. no
case is preferred to any other case.
Permutation
A permutation is an arrangement of all or part of a set of objects. The number of
permutations of n distinct objects taken r at a time is
n
P
r
=
??
n!
nr! ?
Notes on Probability and Statistics
Note: The number of permutations of n distinct objects is n! i.e.,
n
P
n
= n!
The number of permutations of n distinct objects arranged in a circle
is (n ? 1)!
The number of distinct permutations of n things of which n
1
are of one
kind, n
2
of a second kind … n
k
of a k
th
kind is
12 k
n!
n ! n ! ......n !
Combination
A combination is selection of all or part of a set of objects. The number of combinations of
n distinct objects taken r at a time is
??
n
r
n!
C=
r! n r ! ?
Note: In a permutation, the order of arrangement of the objects is important.
Thus abc is a different permutation from bca.
In a combination, the order in which objects are selected does not matter.
Thus abc and bca are the same combination.
Definition of Probability
Consider a random experiment which results in a sample space containing n(S) cases
which are exhaustive, mutually exclusive and equally likely. Suppose, out of n(S) cases,
n(A) cases are favourable to an event A. Then the probability of event A is denoted by
P(A) and is defined as follows.
P(A) =
n(A)
n(S)
=
number of cases favourable to event A
number of cases in the sample space S
Complement of an event
The complement of an event A is denoted by A and it contains all the elements of the
sample space which do not belong to A.
For example: Random experiment: an unbiased die is rolled.
S = {1, 2, 3, 4, 5, 6}
(i) Let A: number on the die is a perfect square
? A = {1, 4} ? A = {2, 3, 5, 6}
(ii) Let B: number on the die is a prime number
? B = {2, 3, 5} ?B = {1, 4, 6}
Note: P(A) + P( A ) = 1 i.e. P(A) = 1 ? P( A )
For any events A and B, ?? ? ? ? ?
?? PA=PAB+PAB
Independent Events
Two events A & B are said to be independent if
P(A ? B) = P(A).P(B)
Note: If A & B are independent then
A & B are independent
A & B are independent
A &B are independent
Theorems of Probability
Addition Theorem
If A and B are any two events then
P(A ?B) = P(A) + P(B) – P(A ?B)
Note: 1. A ? B : either A or B or both i.e. at least one of A & B
AB ? : neither A nor B i.e. none of A & B
A ?B & AB ? are complement to each other
? P(AB ? ) = 1 – P(A ? B)
2. If A & B are mutually exclusive, P(A ? B) = 0
?P(A ? B) = P(A) + P(B)
3. ?? ? ? ? ? ? ? ? ? ?? ?? ? ?
1 2 3 1 231 2
PA A A =P A +PA +PA P A A
? ? ? ? ? ? ??? ? ? ? ?
23 3 1 1 23
PA A P A A +PA A A
Notes on Probability and Statistics
Multiplication Theorem
If A & B are any two events then
P(A ? B) = P(A).P(B/A) = P(B).P(A/B)
1. Conditional probability of occurrence of event B given that event A has already
occurred.
P(B/A) =
? ?
??
PA B
PA
?
2. Conditional probability of occurrence of event A given that event B has already
occurred
P(A/B) =
? ?
??
PA B
PB
?
Bayes’ Theorem
Suppose that a sample space S is a union of mutually disjoint events B
1
, B
2
, B
3
, ..., B
n
,
suppose A is an event in S, and suppose A and all the B
i
’s have nonzero probabilities. If
k is an integer with 1 = k = n, then
P(B
k
/ A) =
? ? ? ?
? ??? ???? ????
kk
11 2 2 n n
PA/B P B
P A / B P B +P A / B P B +...+P A / B P B
Solved Example 1 :
A single die is tossed. Find the probability
of a 2 or 5 turning up.
Solution :
The sample space is S = {1, 2, 3, 4, 5, 6}
P(1) = P(2) = … = P(6) =
1
6
The event that either 2 or 5 turns up in
indicated by 2 ? 5. Thus
? ? ?? ??
11 1
P2 5 P 2 P 5
66 3
?? ? ? ? ?
Solved Example 2 :
A coin is tossed twice. What is the
probability that at least one head occurs ?
Solution :
The sample space is
S = {HH, HT, TH, TT}
Probability of each outcomes = 1/4
Probability of atleast one head occurring is
P(A) =
111 3
44 4 4
? ??
Solved Example 3 :
A die is loaded in such a way that an even
number is twice as likely to occur as an
odd number. If E is the event that a
number less than 4 occurs on a single toss
of the die. Find P(E).
Solution :
S = {1, 2, 3, 4, 5, 6 }
We assign a probability of w to each odd
number and a probability of 2w to each
even number. Since the sum of the
probabilities must be 1, we have 9w = 1 or
w = 1/9.
Hence probabilities of 1/9 and 2/9 are
assigned to each odd and even number
respectively.
? E = {1, 2, 3}
and P(E) =
12 1 4
99 9 9
? ??
Solved Example 4 :
In the above example let A be the event
that an even number turns up and let B be
the event that a number divisible by 3
occurs. Find P(A ? B) and P(A ? B).
Solution :
A = {2, 4, 6} and
B = {3, 6}
We have,
A ? B = {2, 3, 4, 6} and
A ? B = {6}
By assigning a probability of 1/9 to each
odd number and 2/9 to each even number
??
21 2 2 7
PA B
99 9 9 9
?? ? ? ? ?
and ??
2
PA B
9
? ?
Solved Example 5 :
A mixture of candies 6 mints, 4 toffees and
3 chocolates. If a person makes a random
selection of one of these candies, find the
probability of getting (a) a mint, or
(b) a toffee or a chocolate.
Solution :
(a) Since 6 of the 13 candies are mints,
the probability of event M, selecting
mint at random, is
P(M) =
6
13
(b) Since 7 of the 13 candies are toffees
or chocolates it follows that
??
7
PT C
13
??
Solved Example 6 :
In a poker hand consisting of 5 cards, find
the probability of holding 2 aces and
3 jacks.
Solution :
The number of ways of being dealt 2 aces
from 4 is
4
2
4!
C6
2!2!
???
The number of ways of being dealt 3 jacks
from 4 is
4
3
4!
C4
3!1!
? ?
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