Unit 1-probability

UNIT-1:
Points to be covered:
(A) Probability:
• Terminology: random experiment, sample space,
events, types of event
• Definition of probability
• Addition and multiplication theorem
• Conditional theorem, examples.
(B) Mathematical expectation (properties without
proof)

( C) Probability Distribution:
• Meaning, condition, p.d.f., properties and uses
of
• (i) Binomial Distribution
• (ii) Poisson Distribution
• (iii) Normal Distribution.

Terminology
1. Random Experiment:
• An experiment which can result in any one of
the several possible outcomes is called a
random experiment.
1)Tossing of a coin is a random experiment.
2)Drawing a card from a pack of playing card is
a random experiment.
3)Throwing a die is a random experiment.

2. Sample Space
• A set representing all possible outcomes of a
random experiment is called a sample space.
• It is denoted by S or U.
• Each outcome is called a sample point.
• If the number of sample points of S is finite, it
is known as a finite sample space.
• If the number of sample points of S is infinite,
it is known as a infinite sample space.

• For example, if a coin is tossed, the sample
space will be as follows:
S = {H,T}
• Similarly, if two coins are tossed,
S= { HH, HT, TH, TT}

3. Event
• The result of an experiment are known as
event.
• For example,
(1)If a coin is tossed head (H) and tail (T) are
two different events.
(2) 1,2,3,4,5,6 are different events when a dice is
thrown.

4. Exhaustive events:
• If all possible outcomes of an experiment are
considered, the outcomes are said to be
exhaustive.
• The exhaustive events are nothing but all the
sample points in the sample points.
• In throwing a die 1,2,3,4,5,6 are exhaustive
event.

5. Mutually exclusive event:
• Events are said to be mutually exclusive, if
they can not occur together.
• That is, the occurrence of any one of them
prevents the occurrence of the remaining.
• If A and B are two mutually exclusive events,
then A ∩ B = Φ.
• Head and tail are mutually exclusive events
when a coin is tossed.

6. Equally likely events:
• Events are said to be equally likely if we have
no reason to believe that one event is
preferable to the others.
• Head and tail are equally likely events in
tossing a coin.

7. Favorable cases:
• The number of sample points favourable to the
happening of an event A are known as
favourable cases of A.
• For example, in drawing a card from a pack of
cards, the favourable cases for getting a spade
are 13.

8. Independent Events:
• Events are said to be independent if the
happening of one event does not depend upon
the happening or non – happening of the other
events.
For example,
• When a coin is tossed two times, the event of
getting head in the first throw and that of getting
head in the second throw are independent events.

Definition:
• If an experiment can result in n exhaustive,
mutually exclusive and equally likely ways, and
if m of them are favourable to the happening of
an event A, then the probability of happening of
an event A is defined as the ratio of m to n.
• The probability of happening of an event A is
denoted by P(A).
i.e. P(A)= favourable cases / total cases
= m / n

• The probability of an event is always between
0 and 1.
• When P(A) = 0, the event is impossible.
• When P(A) = 1, the event is certain to happen.
• The total probability of happening an event
and not happening an event is 1.
i.e. P(A) + P(A’) = 1.

Addition Theorem of Probability
Statement:
If A and B any two events then,
P(AUB) = P(A) + P(B) – P(A∩B)
Proof:
Two events A and B are shown in Venn-diagram.
From the figure it is clear that the event A
consists of two mutually exclusive events
A∩B’ and A∩B.

A = (A∩B’) U (A∩B)
P(A) = P (A∩B’) + P(A∩B)…………………. (1)
Similarly, the event B consists of two mutually
exclusive events A∩B and A’∩B.
 B = (A∩B) U (A’∩B)
P(B) = P (A∩B) + P(A’∩B)…………………. (2)
From Venn-diagram,
AUB consists of three mutually exclusive events
(A∩B’), (A∩B) and (A’∩B).
(AUB) = (A∩B’) U (A∩B) U (A’∩B)
P(AUB) = P(A∩B’) + P(A∩B) + P(A’∩B)……(3)

Substitute the value of P(A∩B’) and P(A’∩B)
from result (1) and (2) we get
P(AUB) = P(A) - P(A∩B)+ P(B) - P(A∩B) +
P(A∩B)
P(AUB) = P(A) + P(B) - P(A∩B).

RESULTS
1. If A and B are mutually exclusive events (A∩B) = Φ.
Hence P (A∩B) = 0.
P(AUB) = P(A) + P(B) .
2. For three events A, B and C
P(AUBUC) = P(A) + P(B) + P(C) - P(A∩B) - P(B∩C) -
P(A∩C) + P(A∩B ∩C)

Illustration:
1. A bag contains 4 white and 3 black balls. Find the
probability of drawing a white ball from it.
2. Find the probability of getting an odd number when
a cubical die is thrown.
3. Two cards are drawn from a well shuffled pack of
52 cards. Find the probability that both are kings.
4. Two dice are thrown simultaneously. Find the
probability of obtaining total 10 on the two dice.
5. Three coins are thrown simultaneously, find the
probability of getting two heads and one tail.
6. Find the probability of 53 Sundays in a leap year.

Conditional Probability
• Suppose there are 100 persons in a group, 60
males and 40 females.
• Suppose 15 of them use spectacles and out of
them 10 are males.
• If A denotes male and B denotes a person
using spectacles then
P(A) = 60/100 and P(B) = 15/100
• Now, the probability of a person using
spectacles given that he is a male = 10/60.

• This probability of occurrence of an event B
when it is known that A has occurred is known
as conditional probability of B under the
condition that A has occurred and it is denotes
by P (B/A).
• Therefore, P (B/A) = 10/60
• Thus the probability of happening an event B,
when A has happened is defined as conditional
probability of B under A and it is denoted by P
(B/A)

Multiplication Theorem
Statement :
For two events A and B prove that
P(A ∩ B) = P(A) . P (B/A)
OR
P(A ∩ B) = P(B) . P (A/B)
Proof:
Suppose the sample space contains n sample points of
which m are favourable to the occurrence of A and m1
are favourable to the occurrence of A ∩ B.

1 1
1
1
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( ). ( / ) .
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( )
( / )
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,
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( )
m mm
P A P A B P B A
n n m
mm
P A P B A
n m
m
n
P A B
P A P B A P A B
OR
P A B
P B A
P A
similarly
P B P A B P A B
OR
P A B
P A B
P B
    
 

 
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
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

• If A and B are independent events, then
P(A/B) = P(A)
Therefore P(A) . P(B) = P (A ∩ B)
This is known as the multiplication theorem.

ILLUSTRATION
1. Two cards are drawn at random from a pack of 52
cards. Find the probabilities that
(i) One is king and the other is queen.
(ii) Both are spade
(iii)Both are of the same suit.
2. There are 5 red and 7 black balls in an urn. Two
balls are drawn at random one after the other. If
they are drawn (i) with replacement (ii) without
replacement, find the probability that both the
balls are red.
3. If P(A) = 1/3, P(B’) = ¼ and P(A∩ B) = 1/6, find
P(AUB), P (A’ ∩ B’) and P (A’/B’).

1. A machine is made up of two parts A and B.
The probability that part A is defective is 0.05
and the probability that part B is defective is
0.07. find the probability that the entire
machine is not defective.
2. The probability that A speaks the truth is 0.6
and the probability that B speaks the truth is
0.7. They both agree about a statement. Find
the probability that the statement is true.

Mathematical Expectation
(A) Discrete Random Variable:
• If two coins are tossed simultaneously the sample
space S = {HH, HT, TH, TT}
• Suppose we are interested in number of heads
obtained.
• If ‘x’ denotes the number of heads the values of ‘x’
can be 0, 1,2.
• Here ‘x’ is said to be a discrete random variable.
• A variable whose values can be obtained from the
results of a random experiment is called a random
variable.

• A random variable is usually denoted by ‘x’.
• If a random variable can take finite values or
countable infinite values it is known as a
discrete variable.
• The number of children in a family, number of
accidents are examples of discrete variables.

(B) Mathematical expectation of a discrete
variable:
• The expected value of a discrete random variable
is the average value of the random variable.
• If x1, x2,….xn are different values of a random
variable ‘x’ with their respective probabilities
p(x1), p(x2),…… p(xn) then the expected value of
the random variable ‘x’ is defined as follows:
E(x) = x1.p(x1) + x2.p(x2) + …… xnp(xn)
= ∑xi p(xi)
• The expected value of ‘x’ is the mean of ‘x’ and is
denoted by μ.
• Therefore, E(x) = μ

Properties of expected value:
1. The expected value of a constant is a constant
itself. i.e. E(k) = k.
2. If k is a constant, E(kx) = k E(x).
3. E(ax + b) = a E(x) + b.
4. If x and y are two random variables,
E(x + y ) = E (x) + E (y).
5. if x and y are two independent random
variables,
E (xy) = E(x).E(y)

Illustration
1. An unbiased die is thrown. Find the expected
value of the number on the die.
2. Two dice are thrown. Find the expected value
of the total on the dice.

(B) Variance of a random variable:
• The average of the squares of the deviations from the
mean of a random variable x is said to be its variance.
• If the mean of a random variable x is E(x) = μ the
variance of x can be given as follows:
V(x) = E (x – μ )2
= E (x2 – 2x μ + μ2)
= E (x2) - 2 μ E(x) + E (μ2)
= E (x2) - 2 μ μ + μ 2
= E(x2) - 2 μ 2 + μ 2
= E(x2) – μ2
= E(x2) – [E(x)]2
V(x) = E(x2) - [E(x)]2

Illustration:
1. A person throws an unbiased die and he gets
the amount in rupees equal to the square of
the face value obtained. Find the
mathematical expectation of his amount.
2. Discrete random variable x has prob.
Distribution as under. Then find (i) E (4x + 6)
(ii) V (2x + 3)
X 0 1 2 3 4 5
P(x) 1/3 ¼ 1/12 1/8 1/6 1/24

Probability Distribution
• The arrangement of different values of a discrete
variable and the probabilities associated with
these values is known as a discrete probability
distribution.
• The study of the distribution of a single variable
is called the study of a univariate distribution.

(A) Binomial Distribution:
• Binomial distribution was first given by a Swiss
mathematician James Bernoulli in the
beginning of 18th century.
• When an experiment can result only in two
ways success and failure, and when such an
experiment can be repeated independently and
in each trial probability of success remains the
same, then such trials are known as Bernoulli
trials.
• Tossing of a coin for ‘n’ times is an example of
Bernoulli trials.

Assumption
1. The experiment in which only two results
success or failure are obtained.
2. The experiment can be repeated
independently for a fixed number of times
‘n’.
3. The probability of success p remains same in
each trial.

Probability distribution function (p.d.f.)
• Probability distribution function of Binomial
distribution is
• Where n= total no. of trials.
x= probability of getting success out of
n independent success.
p = probability of success
q = probability of failure.
( ) , 0,1,2,......., .n x n x
xp x C p q x n
 

Properties of Binomial distn.
1. This is distribution of discrete variable.
2. ‘n’ and ‘p’ are parameters of B.D.
3. Mean of B.D. is np which shows the average number of
success.
4. Variance of B.D. is npq. And S.D. = √npq.
5. When p and q are equal i.e. p = q= ½, B.D. is a
symmetrical distribution.
6. When p < ½, its skewness is positive and when p > ½,
its skewness is negative.
7. Variance is always less than its mean.
8. When n is very large and p or q is very small, Binomial
distribution tend to Poisson Distribution
9. When n is very large and p or q are not very small,
Binomial distribution tend to Normal Distribution

Uses
1. Finding out probability of number of
successes out of n independent Bernoulli
trials.
2. Finding control limits of p and np charts in
Statistical Quality Control.
3. In large sample tests for attributes.

(B) Poisson Distribution:
• It is distribution of a discrete variable.
• It was first given by French mathematician
Simeon De Poisson in 1837, as a limiting case
of Binomial distribution.
• When n is very large, p is very small and np is
constant, then it tend to Poisson distribution.

p.d.f
• Probability distribution function of Poisson
distribution is given by
• Probability of x successes P(x) =
where e = 2.7183
m = mean = np
x = no. of successes.
!
m x
e m
x


Application of P.D.
1. Number of accidents on a road.
2. Number of misprint per page of a book.
3. Number of defects in a radio set.
4. Number of air bubbles in a glass bottle.
5. Number of suicides committed per day.
6. Number of goals scored in a football match.
7. Number of telephone calls received during a
given interval of time.
Thus, Poisson distribution is a distribution of
rare events.

Properties:
1. This is distribution of discrete variable.
2. It is distribution of rare occurrence.
3. ‘m’ is parameter of the distribution.
4. Mean of the distribution is ‘m’.
5. Variance of distribution is ‘m’.
6. Sum of two independent Poisson variate is
also a Poisson variate.
7. This is a distribution with positive skewness.

(C ) Normal Distribution
• This distribution was first given by De Movire in
1773.
• He developed this distribution as a limiting case of
Binomial distribution.
• In binomial distribution when n is very large and p
and q are not very small, it tend to normal
distribution.
• Gauss and Laplace also derived the mathematical
form of this distribution from the study of
distribution of errors.
• The graph of normal distribution called normal
curve, is bell shaped and it extends infinitely on both
the sides of x axis, but does not meet it.

p.d.f of Normal Distribution:
• If a continuous random variable x obeys the
following probability law, it is said to be
normally distributed.
• Here μ and σ are mean and S.D. of the
distribution.
• They are called the parameters of the
distribution.
2
1
21
( ) ;
2
x
f x e x



 
  
 
    


Properties of Normal Distribution
1. This is a distribution of a continuous variable.
2. μ and σ are the parameters of this distribution.
3. The curve of the normal distribution is
symmetrical about mean and it is bell shaped.
4. Mean, median and mode are equal in this
distribution.
5. Its skewness is zero.
6. The tails of the normal curve do not meet x axis.
7. Mean deviation about mean = (4/5) σ (approx.)
8. Quartile deviation = (2/3) σ (approx.)

Importance of normal distribution
1. It is used in the field of Sociology, Psychology,
Economics etc.
2. In SQC, the properties of normal distribution
are used.
3. In significant tests for large samples, the
distribution is widely used.
4. It is used in small sample tests in which it is
assumed that the population from which the
sample is drawn is normal.

Unit 1-probability

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Unit 1-probability