Mathematics

Probability
Table of Content

1. Introduction.
2. Definitions of various terms.
3. Classical definition of probability.
4. Some important remarks about coins, dice, playing cards.
5. Problems based on combination and permutation.
6. Odds in favor and odds against an event.
7. Addition theorems on probability.
8. Conditional probability.
9. Total probability and Baye's rule.
10. Binomial distribution.

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1. Introduction.

Numerical study of chances of occurrence of events is dealt in probability theory.

The theory of probability is applied in many diverse fields and the flexibility of the theory provides
approximate tools for so great a variety of needs.
There are two approaches to probability viz. (i) Classical approach and (ii) Axiomatic approach.
In both the approaches we use the term ‘experiment’, which means an operation which can produce
some well-defined outcome(s). There are two types of experiments:

(1) Deterministic experiment: Those experiments which when repeated under identical conditions
produce the same result or outcome are known as deterministic experiments. When experiments in
science or engineering are repeated under identical conditions, we get almost the same result everytime.

(2) Random experiment: If an experiment, when repeated under identical conditions, do not produce
the same outcome every time but the outcome in a trial is one of the several possible outcomes then
such an experiment is known as a probabilistic experiment or a random experiment.
In a random experiment, all the outcomes are known in advance but the exact outcome is unpredictable.
For example, in tossing of a coin, it is known that either a head or a tail will occur but one is not sure if a
head or a tail will be obtained. So it is a random experiment.

2. Definitions of Various Terms.

(1) Sample space: The set of all possible outcomes of a trial (random experiment) is called its sample
space. It is generally denoted by S and each outcome of the trial is said to be a sample point.

Example: (i) If a dice is thrown once, then its sample space is S = {1, 2, 3, 4, 5, 6}
(ii) If two coins are tossed together then its sample space is S = {HT, TH, HH, TT}.

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(2) Event: An event is a subset of a sample space.

(i) Simple event: An event containing only a single sample point is called an elementary or simple event.
Example: In a single toss of coin, the event of getting a head is a simple event.
Here S = {H, T} and E = {H}

(ii) Compound events: Events obtained by combining together two or more elementary events are
known as the compound events or decomposable events.
For example, In a single throw of a pair of dice the event of getting a doublet, is a compound event
because this event occurs if any one of the elementary events (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) occurs.

(iii) Equally likely events: Events are equally likely if there is no reason for an event to occur in
preference to any other event.
Example: If an unbiased die is rolled, then each outcome is equally likely to happen i.e., all elementary
events are equally likely.

(iv) Mutually exclusive or disjoint events: Events are said to be mutually exclusive or disjoint or
incompatible if the occurrence of any one of them prevents the occurrence of all the others.
Example: E = getting an even number, F = getting an odd number, these two events are mutually
exclusive, because, if E occurs we say that the number obtained is even and so it cannot be odd i.e., F
does not occur.
A1 and A2 are mutually exclusive events if A1  A 2   .

(v) Mutually non-exclusive events: The events which are not mutually exclusive are known as
compatible events or mutually nonexclusive events.

(vi) Independent events: Events are said to be independent if the happening (or non-happening) of one
event is not affected by the happening (or non-happening) of others.
Example: If two dice are thrown together, then getting an even number on first is independent to getting
an odd number on the second.

(vii) Dependent events: Two or more events are said to be dependent if the happening of one event
affects (partially or totally) other event.
Example: Suppose a bag contains 5 white and 4 black balls. Two balls are drawn one by one. Then two
events that the first ball is white and second ball is black are independent if the first ball is replaced

3
before drawing the second ball. If the first ball is not replaced then these two events will be dependent
because second draw will have only 8 exhaustive cases.

(3) Exhaustive number of cases: The total number of possible outcomes of a random experiment in a
trial is known as the exhaustive number of cases.
Example : In throwing a die the exhaustive number of cases is 6, since any one of the six faces marked
with 1, 2, 3, 4, 5, 6 may come uppermost.

(4) Favourable number of cases: The number of cases favourable to an event in a trial is the total
number of elementary events such that the occurrence of any one of them ensures the happening of the
event.
Example : In drawing two cards from a pack of 52 cards, the number of cases favourable to drawing 2 queens is
4
C2 .

(5) Mutually exclusive and exhaustive system of events: Let S be the sample space associated with a
random experiment. Let A1, A2, …..An be subsets of S such that
(i) Ai  A j   for i  j and (ii) A1  A 2  ....  An  S

Then the collection of events A1 , A 2 , ....., An is said to form a mutually exclusive and exhaustive system of
events.
If E1 , E 2 , ....., En are elementary events associated with a random experiment, then
(i) Ei  E j   for i  j and (ii) E1  E 2  ....  En  S

So, the collection of elementary events associated with a random experiment always form a system of
mutually exclusive and exhaustive system of events.
In this system, P( A1  A 2 .......  An )  P( A1 )  P( A 2 )  .....  P( An )  1 .

Important Tips
 Independent events are always taken from different experiments, while mutually exclusive events
are taken from a single experiment.
 Independent events can happen together while mutually exclusive events cannot happen
together.
 Independent events are connected by the word “and” but mutually exclusive events are
connected by the word “or”.

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3. Classical definition of Probability.

If a random experiment results in n mutually exclusive, equally likely and exhaustive outcomes, out of
which m are favorable to the occurrence of an event A, then the probability of occurrence of A is given
by
m Number of outcomes favourable to A
P( A)  
n Number of totaloutcomes
It is obvious that 0 ≤ m ≤ n. If an event A is certain to happen, then m = n, thus P(A) = 1.
If A is impossible to happen, then m = 0 and so P(A) = 0. Hence we conclude that
0 ≤ P(A) ≤ 1.
Further, if A denotes negative of A i.e. event that A doesn’t happen, then for above cases m, n; we shall
have
n m m
P( A )   1   1  P( A)
n n
 P( A)  P( A )  1 .

Notations: For two events A and B,

(i) A or A or AC stands for the non-occurrence or negation of A.
(ii) A  B stands for the occurrence of at least one of A and B.
(iii) A  B stands for the simultaneous occurrence of A and B.
(iv) A  B stands for the non-occurrence of both A and B.
(v) A  B stands for “the occurrence of A implies occurrence of B”.

4. Some important remarks about Coins, Dice, Playing cards and Envelopes.

(1) Coins: A coin has a head side and a tail side. If an experiment consists of more than a coin, then coins
are considered to be distinct if not otherwise stated.
Number of exhaustive cases of tossing n coins simultaneously (or of tossing a coin n times) = 2n.

(2) Dice: A die (cubical) has six faces marked 1, 2, 3, 4, 5, 6. We may have tetrahedral (having four faces 1,
2, 3, 4) or pentagonal (having five faces 1, 2, 3, 4, 5) die. As in the case of coins, if we have more than one
die, then all dice are considered to be distinct if not otherwise stated.
Number of exhaustive cases of throwing n dice simultaneously (or throwing one dice n times) = 6n.
(3) Playing cards: A pack of playing cards usually has 52 cards. There are 4 suits (Spade, Heart, Diamond
and Club) each having 13 cards. There are two colours red (Heart and Diamond) and black (Spade and
Club) each having 26 cards.

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In thirteen cards of each suit, there are 3 face cards or coart cards namely king, queen and jack. So there
are in all 12 face cards (4 kings, 4 queens and 4 jacks). Also there are 16 honour cards, 4 of each suit
namely ace, king, queen and jack.

(4) Probability regarding n letters and their envelopes: If n letters corresponding to n envelopes are
placed in the envelopes at random, then
1
(i) Probability that all letters are in right envelopes  .
n!
1
(ii) Probability that all letters are not in right envelopes  1  .
n!
1 1 1 1
(iii) Probability that no letter is in right envelopes     ...  (1)n .
2! 3! 4 ! n!
1 1 1 1 1 
(iv) Probability that exactly r letters are in right envelopes      .....  (1)n  r .
r!  2! 3! 4 ! (n  r)!

5. Problems based on Combination and Permutation.

n!
(1) Problems based on combination or selection: To solve such kind of problems, we use n C r  .
r!(n  r)!

(2) Problems based on permutation or arrangement: To solve such kind of problems, we use
n!
n
Pr  .
(n  r)!

6. Odds In favor and Odds against an Event.

As a result of an experiment if “a” of the outcomes are favorable to an event E and “b” of the outcomes
are against it, then we say that odds are a to b in favor of E or odds are b to a against E.
Number of favourable cases a a /(a  b) P(E)
Thus odds in favour of an event E     .
Number of unfavourab le cases b b /(a  b) P(E )
Number of unfavourab le cases b P(E )
Similarly, odds against an event E    .
Number of favourable cases a P(E)

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Important Tips
 If odds in favour of an event are a : b, then the probability of the occurrence of that event is
a b
and the probability of non-occurrence of that event is .
ab ab
b
 If odds against an event are a : b, then the probability of the occurrence of that event is
ab
a
and the probability of non-occurrence of that event is .
ab

7. Addition Theorems on Probability.

Notations: (i) P( A  B) or P( A  B) = Probability of happening of A or B

= Probability of happening of the events A or B or both
= Probability of occurrence of at least one event A or B
(ii) P(AB) or P(AB) = Probability of happening of events A and B together.

(1) When events are not mutually exclusive: If A and B are two events which are not mutually
exclusive, then P( A  B)  P( A)  P(B)  P( A  B) or P( A  B)  P( A)  P(B)  P( AB) .

For any three events A, B, C

P( A  B  C)  P( A)  P(B)  P(C)  P( A  B)  P(B  C)  P(C  A)  P( A  B  C)
or P( A  B  C)  P( A)  P(B)  P(C)  P( AB)  P(BC)  P(CA )  P( ABC ) .

(2) When events are mutually exclusive: If A and B are mutually exclusive events, then
n( A  B)  0  P( A  B)  0
 P( A  B)  P( A)  P(B) .
For any three events A, B, C which are mutually exclusive,
P( A  B)  P(B  C)  P(C  A)  P( A  B  C) = 0  P( A  B  C)  P( A)  P(B)  P(C) .
The probability of happening of any one of several mutually exclusive events is equal to the sum of their
probabilities, i.e. if A1 , A 2 ..... An are mutually exclusive events, then

 A )   P(A ) .
P( A1  A 2  ...  An )  P( A1 )  P( A 2 )  .....  P( An ) i.e. P( i i

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(3) When events are independent: If A and B are independent events, then P( A  B)  P( A).P(B)
 P( A  B)  P( A)  P(B)  P( A).P(B) .

(4) Some other theorems

(i) Let A and B be two events associated with a random experiment, then
(a) P( A  B)  P(B)  P( A  B) (b) P( A  B )  P( A)  P( A  B)

If B  A, then
(a) P( A  B )  P( A)  P(B) (b) P(B)  P( A)

Similarly if A  B, then
(a) ( A  B)  P(B)  P( A) (b) P( A)  P(B) .

Note: Probability of occurrence of neither A nor B is P( A  B )  P( A  B)  1  P( A  B) .

(ii) Generalization of the addition theorem: If A1 , A 2 ,....., An are n events associated with a random
experiment, then
 n  n n n
P  Ai  
  P( A )   P( A
i i  Aj)   P( A i  A j  Ak )  ...  (1)n 1 P( A1  A 2  .....  An ) .
 i1  i1 i, j1 i, j, k 1
i j i jk

 n  n
If all the events Ai (i  1, 2..., n) are mutually exclusive, then P  Ai     P( A ) i
 i1  i1

i.e. P( A1  A 2  ....  An )  P( A1 )  P( A 2 )  ....  P( An ) .

(iii) Booley’s inequality: If A1 , A 2 , .... An are n events associated with a random experiment, then
 n  n  n  n
(a) P  Ai  
  P( A )  (n  1)
i (b) P  A i  
  P( A ) i
 i1  i1  i1  i1

These results can be easily established by using the Principle of Mathematical Induction.
Important Tips
Let A, B, and C are three arbitrary events. Then

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 Verbal description of event Equivalent Set Theoretic Notation
(i) Only A occurs (i) A  B  C
(ii) Both A and B, but not C occur (ii) A  B  C
(iii) All the three events occur (iii) A  B  C
(iv) At least one occurs (iv) A  B  C
(v) At least two occur (v) ( A  B)  (B  C)  (A  C)
(vi) One and no more occurs (vi) ( A  B  C ) ( A  B  C )  ( A  B  C)
(vii) Exactly two of A, B and C occur (vii) ( A  B  C ) ( A  B  C)  (A  B  C)
(viii) None occurs (viii) A  B  C  A  B  C
(ix) Not more than two occur (ix) (A  B)  (B  C)  (A  C)  ( A  B  C)
(x) Exactly one of A and B occurs (x) ( A  B )  ( A  B)

8. Conditional Probability.

Let A and B be two events associated with a random experiment. Then, the probability of occurrence of
A under the condition that B has already occurred and P(B)  0, is called the conditional probability and
it is denoted by P(A/B).
Thus, P(A/B) = Probability of occurrence of A, given that B has already happened.
P( A  B) n( A  B)
  .
P ( B) n(B)

Similarly, P(B/A) = Probability of occurrence of B, given that A has already happened.

P( A  B) n( A  B)
  .
P( A) n( A)

Note: Sometimes, P(A/B) is also used to denote the probability of occurrence of A when B occurs. Similarly, P(B/A)
is used to denote the probability of occurrence of B when A occurs.

(1) Multiplication theorems on probability

(i) If A and B are two events associated with a random experiment, then P( A  B)  P( A) . P(B / A) , if P(A)
 0 or P( A  B)  P(B) . P( A / B) , if P(B)  0.

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(ii) Extension of multiplication theorem: If A1 , A 2 , ...., An are n events related to a random experiment,
then P( A1  A 2  A3  ....  An )  P( A1 )P( A 2 / A1 )P( A3 / A1  A 2 )....P( An / A1  A 2  ...  An1 ) ,
Where P( Ai / A1  A 2  ...  Ai1 ) represents the conditional probability of the event A i , given that the
events A1 , A 2 , ....., Ai1 have already happened.

(iii) Multiplication theorems for independent events: If A and B are independent events associated
with a random experiment, then P( A  B)  P( A) . P(B) i.e., the probability of simultaneous occurrence of
two independent events is equal to the product of their probabilities.
By multiplication theorem, we have P( A  B)  P( A) . P(B / A) .
Since A and B are independent events, therefore P(B / A)  P(B) . Hence, P( A  B)  P( A) . P(B) .

(iv) Extension of multiplication theorem for independent events: If A1 , A 2 , ...., An are independent
events associated with a random experiment, then P( A1  A 2  A3  ...  An )  P( A1 )P( A 2 )...P( An ) .
By multiplication theorem, we have
P( A1  A 2  A3  ...  An )  P( A1 )P( A 2 / A1 )P( A3 / A1  A 2 )...P( An / A1  A 2  ...  An 1 )
Since A1 , A 2 , ...., An 1 , An are independent events, therefore
P( A 2 / A1 )  P( A 2 ), P( A3 / A1  A 2 )  P( A3 ),...., P( An / A1  A 2  ...  An 1 )  P( An )
Hence, P( A1  A2  ...  An )  P( A1 )P( A2 )....P( An ) .

(2) Probability of at least one of the n independent events: If p1 , p 2 , p 3 , ........, p n be the probabilities
of happening of n independent events A1 , A 2 , A3 , ........, An respectively, then

(i) Probability of happening none of them

 P( A1  A2  A3 ......  An )  P( A1 ).P( A2 ).P( A3 ).....P( An )  (1  p1 )(1  p 2 )(1  p 3 )....(1  p n ) .

(ii) Probability of happening at least one of them

 P( A1  A2  A3 ....  An )  1  P( A1 )P( A2 )P( A3 )....P( An )  1  (1  p1 )(1  p 2 )(1  p 3 )...(1  p n ) .

(iii) Probability of happening of first event and not happening of the remaining
 P( A1 )P( A2 )P( A3 ).....P( An )  p1 (1  p 2 )(1  p 3 ).......(1  pn )

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9. Total Probability and Baye’s rule.

(1) The law of total probability: Let S be the sample space and let E1 , E 2 , ..... En be n mutually exclusive
and exhaustive events associated with a random experiment. If A is any event which occurs with E1 or
E 2 or …. or En, then P( A)  P(E1 ) P( A / E1 )  P(E 2 ) P( A / E 2 )  ...  P(En ) P( A / En ) .

(2) Baye’s rule: Let S be a sample space and E1 , E 2 , ..... En be n mutually exclusive events such that
n

E i  S and P(Ei )  0 for i = 1, 2, ……, n. We can think of (Ei’s as the causes that lead to the outcome
i1

of an experiment. The probabilities P(Ei), i = 1, 2, ….., n are called prior probabilities. Suppose the
experiment results in an outcome of event A, where P(A) > 0. We have to find the probability that the
observed event A was due to cause Ei, that is, we seek the conditional probability P(Ei / A) . These
P(E i ).P( A / E i )
probabilities are called posterior probabilities, given by Baye’s rule as P(E i / A)  n
.
 P( E
k 1
k ) P( A / Ek )

10. Binomial Distribution.

(1) Geometrical method for probability: When the number of points in the sample space is infinite, it
becomes difficult to apply classical definition of probability. For instance, if we are interested to find the
probability that a point selected at random from the interval [1, 6] lies either in the interval [1, 2] or [5, 6],
we cannot apply the classical definition of probability. In this case we define the probability as follows:
Measure of region A
P{x  A}  ,
Measure of the sample space S

Where measure stands for length, area or volume depending upon whether S is a one-dimensional, two-
dimensional or three-dimensional region.

(2) Probability distribution: Let S be a sample space. A random variable X is a function from the set S to
R, the set of real numbers.
{11, 12, , 16
21, 22, , 26
For example, the sample space for a throw of a pair of dice is S 
   
61, 62, , 66}

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Let X be the sum of numbers on the dice. Then X (12)  3, X (43)  7 , etc. Also, {X = 7} is the event {61, 52,
43, 34, 25, 16}. In general, if X is a random variable defined on the sample space S and r is a real number,
then {X = r} is an event. If the random variable X takes n distinct values x 1 , x 2 , ...., x n , then { X  x 1 } ,
{X  x 2 }, ...., {X  x n } are mutually exclusive and exhaustive events.

X = x3 S
X = x1
X = x2

X = x4
X = xn

Now, since (X  x i ) is an event, we can talk of P(X  x i ) . If P(X  x i )  Pi (1  i  n) , then the system of
numbers.
 x1 x2  xn 
 
 p1 p2  p n 
is said to be the probability distribution of the random variable X. The expectation (mean) of the random
n
variable X is defined as E( X )  p x
i1
i i

n n
and the variance of X is defined as var(X )  i1
p i (x i  E( X ))2  p x
i1
i
2
i  (E( X ))2 .

(3) Binomial probability distribution: A random variable X which takes values 0, 1, 2, …, n is said to
follow binomial distribution if its probability distribution function is given by
P(X  r)  n Cr p r q nr , r  0, 1, 2, ....., n
Where p, q > 0 such that p + q = 1.
The notation X ~ B(n, p) is generally used to denote that the random variable X follows binomial
distribution with parameters n and p.
We have P(X  0)  P(X  1)  ...  P(X  n)  n C0 p 0 q n0  n C1 p 1 q n1  ...  n Cn p n q nn  (q  p)n  1n  1 .

Now probability of
(a) Occurrence of the event exactly r times
P(X  r)  n C r q n r p r .
(b) Occurrence of the event at least r times
n
P( X  r)  n C r q n r p r  ...  p n  
X r
n
C X p X q n X .

(c) Occurrence of the event at the most r times

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 r
P(0  X  r)  q n  n C 1 q n 1 p  ...  n C r q n r p r  p
X 0
X
q n X .

(iv) If the probability of happening of an event in one trial be p, then the probability of successive
happening of that event in r trials is p r .

Note: If n trials constitute an experiment and the experiment is repeated N times, then the frequencies of 0, 1, 2, …,
n successes are given by N .P(X  0), N .P(X  1), N .P(X  2),...., N .P(X  n) .

(i) Mean and variance of the binomial distribution: The binomial probability distribution is
X 0 1 2 n
P( X ) n
C0 q n p 0 n
C1q n 1 p n
C2 q n  2 p 2 ..... n Cn q 0 p n

n n
The mean of this distribution is i1
X i pi   X.
X 1
n
C X q n  X p X  np ,

the variance of the Binomial distribution is  2  npq and the standard deviation is   (npq) .
(ii) Use of multinomial expansion : If a die has m faces marked with the numbers 1, 2, 3, ….m and if
such n dice are thrown, then the probability that the sum of the numbers exhibited on the upper faces
(x  x 2  x 3  ....  x m )n
equal to p is given by the coefficient of x p in the expansion of .
mn

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