Binomial distribution

Contents
Introduction- Definition1
Obtaining Coefficients of Binomial2
Pascal's Triangle3
Properties & Importance4

Introduction
The Binomial Distribution is also known as
Bernoulli Distribution.
Associated with the name of a Swiss
Mathematician James Bernoulli (1654-1705).
Binomial Distribution is a probability
distribution expressing the probability of
dichotonomous alternatives.i.e., Success or
Failure.
Bernoulli process is usually referred to as
Bernoulli trial
Bernoulli Distribution

Assumptions
An experiment is performed under the same
condition for a fixed number of trials, say n
The probability of a success denoted by p remains
constant from trial to trial.
The Probability of a failure denoted by q is equal to
(1-p)
1
2
3
For each trial there are only two possible outcomes
of the experiment. S= {failure, success}.
The trials are statistically significant.

Binomial Distribution
01
04
03
02
If two coins are tossed
simultaneously there are four
possible outcomes:
TT, TH, HT and HH
The probabilities corresponding to these
results are:
TT TH HT HH
qq qp pq pp
q2 2qp p2
These are the terms of the
binomial (q+p)2 because
(q+p)2 = q2+2qp+p2
If a coin is tossed once there are
two outcomes, namely head or tail
The probability of obtaining a head
or p=1/2 and the probability of
obtaining a tail or q = 1/2. Thus
(q+p=1). These are the terms of the
binomial.

wherenp=q=1/2, we have
[1/2+1/2]3 = 1/8+3/8+3/8+1/8
If three coins A, B and C are tossed
the following are the possible
outcomes and the probabilities
corresponding to these results are:
These are the terms of the binomial
(q+p)3
(q+p)3
=q3+3q2p+3qp2+p3
ABC ABC ABC ABC ABC ABC ABC ABC
TTT TTH THT HTT THH HTH HHT HHH
q3 q2p q2p q2p qp2 qp2 qp2 p3

Probability table
for number of
Heads
Numb
er of
Heads
(X)
Probability
P(X=r)
0 nC0p0qn=qn
1 nC1qn-1p1
2 nC2qn-2p2
3 nC3qn-3p3
.
.
.
.
.
.
r nCrqn-rpr
.
.
.
.
.
.
n nCnq0pn =pn
The Binomial Distribution:
p(r)=nCrqn-rpr
Where,
p= Probability of success in a single
trial
q =1-p
n= number of trials
r= number of successes in n trials

Obtaining Coefficients of the Binomial - To find the terms of the expansion of (q+p)n
1 6
43
2 RULES
The first term is qn
When we expand (q+p)n, we get
(q+p)n = qn+nC1qn-1p+nC2qn-2p2+....+nCrqn-rpr+...pn
Where 1, nC1, nC2 .... are binomial coefficients
The Second term is nC1qn-1p Divide the products do obtained by
one more than the power of p in
that preceeding term
In each succeeding term the power of
q is reduced by 1 and the power of p
is increased by 1
The coefficient of any term is found by
multiplying the coefficient of the
preceeding term by the power of q in that
preceeding term
5

PASCAL'S
TRIANGLE
[SHOWING THE COEFFICIENTS OF THE TERM
(q+p)n

Properties of Binomial Distribution
5
4
1
2
3
Thus the mean of binomial
distribution is np.
The standard deviation of
binomial distribution is:
The shape and location of
Binomial Distribution changes as p
changes for a given n or changes
for a given p
As n increases for a fixed p, the binomial distribution
moves to the right, flattens and spreads out. The
mean of the binomial distribution, np increases as n
increases with p held constant
As p increases for a fixed n the
binomial distribution shifts to the
right.
The mode of the binomial
distribution is equal to the
value of x which has the
largest probability. For fixed n,
both mean and mode increase
as p increases.
If n is large and if neither p nor
q is too close to zero, the
binomial distribution can be
closely approximated by a
normal distribution with
standaradized variable given
by:
Z=X=np/√npq
√npq

Binomial
Probability
Binomial Probability Formula:
P(r) = nCr pr qn-r
where, P (r) = Probability of r successes in n trials;
p = Probability of success;
q = Probability of failure = 1-p;
r = No. of successes desired; and
n = No. of trials undertaken.
The determining equation for nCr can easily be written as:
Hence the following form of the equations, for carrying out computations of the
binomial probability is perhaps more convenient.

Binomial
Probability
Example 1
A fair coin is tossed six times. What is the probability of obtaining four or more heads?
Solution: When a fair coin is tossed, the probabilities of head and tail in case of an unbiased
coin are equal, i.e.,

Binomial
Probability
Example 1

Binomial
Probability
Example 2

Fitting a Binomial Distribution
1
2
3

Keyword
s
It is a type of discrete
probability distribution
function that includes an
event that has only two
outcomes (success or
failure) and all the trials are
mutually independent.
Continuous Probability
Distribution
In this distribution the
variable under
consideration can take
any value within a given
range.
Continuous Random
Variable
If the random variable is
allowed to take any
value within a given
range, it is termed as
continuous random
variable
Discrete Probability
Distribution
A probability distribution
in which the variable is
allowed to take on only a
limited number of values.

Self Assessment
The incidence of a certain disease is such that on an average 20% of
workers
suffer from it. If 10 workers are selected at random, find the probability
that
i) Exactly 2 workers suffer from the disease
ii) Not more than 2 workers suffer from the disease
iii) At least 2 workers suffer from the disease

Thank You
Dr.Saradha A
Assistant Professor
Department of Economics
Ethiraj College for Women
Chennai

Binomial distribution

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