The document defines and explains the binomial distribution. It was developed by James Bernoulli in 1700 and describes the probability of success in a fixed number of binary trials where the probability of success is the same for each trial. The key points are: - It gives the probability of x successes in n trials where the probability of success p is constant for each trial. - The formula to calculate binomial probability is P(X) = nCx * px * q(n-x) where n is trials, x is successes, p is probability of success, and q is probability of failure (1 - p). - Important statistics of the binomial distribution are the mean (np), variance (npq),

1. By
Dr. Satyanarayan Pandey
Department of Management Studies
BBMKU, Dhanbad

2. Introduction
 Binomial distribution was given by Swiss
mathematician James Bernouli (1654-1705) in 1700 and
it was first published in 1713. It is also known as
‘Bernouli Distribution’.

3. DEFINITION
 Binomial distribution is a discrete probability
distribution which is obtained when the probability p
of the happening of an event is same in all the trials,
and there are only two events in each trial.
 E.g... The probability of getting a head, when a coin is
tossed a number of times, must remains same in each
toss i.e. P= 1/2.

4. CALCULATION OF BINOMIAL
DISTRIBUTION
 It is a discrete probability distribution. Binomial
Probability is calculated by following general formula-
P(X) = nCx Px q(n-x)
 Where, n = number of trials
x = number of success
p = Probability of success
q = Probability of failure = 1 – p

5. CHARACTERISTICS OF BINOMIAL
DISTRIBUTION
 It is a discrete distribution which gives the theoretical
probabilities.
 It depends on the parameter p or q, the probability of
success or failure and n(i.e. The number of trials). The
parameter n is always a positive integer.
 The distribution will be symmetrical if p=q. It is skew
symmetric or asymmetric if p is not equal to q.

6. STATISTICS OF THE BINOMIAL
DISTRIBUTION
 The statistics of the binomial distribution are:
Mean=np,
 Variance=npq, and
 Standard deviation = √npq
 The mode of the binomial distribution is equal to that
value of x which has longer frequency.

7. CONDITIONS FOR BINOMIAL
DISTRIBUTION
 The random experiment is performed repeatedly a
finite and fixed number of times.
 The outcome of the random experiment(trials) results
in the dichotomous classification of events.
 All the trials are independent.
 The probability of success in any trial is p and is
constant for each trial.
 q= 1-p is then termed as the probability of failure and
is constant for each trial.

8. Example
 E.g... If we toss a fair coin n times (which is fixed and
finite), then the outcome of any trial is one of the
mutually exclusive events, viz, head(success) and
tail(failure). Further, all the trials are independent,
since the result of any throw of coin does not affect
and is not affected by the result of other throws.

9. Thank You