Binomial distribution

BINOMIAL DISTRIBUTION
MEANING & FEATURES

ASSUMPTIONS , PROPERTIES
& FORMULAS

 The prefix ‘Bi’ means two or twice.
 A binomial distribution can be understood as the
probability of a trail with two and only two outcomes.
 It is a discrete distribution.
 The binomial distribution was discovered by Swiss
mathematician James Bernoulli in 1700.
 It is also known as Bernoulli Distribution or Bernoulli
Theorem.

When only two outcomes are
possible, binomial distribution is prepared –
example – throw of coins – (Head or tail) trials
are independent
P= probability of success .
Q= probability of failure .

It is also known as Bernoulli Distribution or
Bernoulli Theorem.
This theorem was published eight years after
the death of James Bernoulli in 1713.
Finite number of trials – with exclusive and
exhaustive outcomes .

BINOMIAL DISTRIBUTION EXAMPLES
If you are purchasing a lottery then either you are
going to win money or you are not. In other
words, anywhere the outcome could be a success
or a failure that can be proved through binomial
distribution.
if someone tosses the coin then there is an equal
chance of outcome it can be heads or tails. There
is a 50% chance of the outcomes.

Examples of election polls; whether the party
‘A’ will win or the party ‘B’ will win in the
upcoming election. Whether by implementing
a certain policy the government will get the
expected results within a specific period or
not.

if you are appearing in an exam then there is also
an equal possibility of getting passed or fail.
The binomial distribution summarized the
number of trials, survey or experiment
conducted.
It is very useful when each outcome has the
equal chance of attaining a particular value.

FEATURES OF BINOMIAL DISTRIBUTION
It is used in such situation where an experiments
results in two possibilities i.e. success or failure.
Mutually Exclusive Outcomes .
All the trials are independent.
The Probability of success in each trial is
constant.

ASSUMPTIONS OF BERNOULLI DISTRIBUTION
1. Three distribution can be applied only in
those situation where the events are mutually
exclusive in each trial with two options only :-
Success & Failure .
2. Size of sample must be finite ,i.e.-fixed.

ASSUMPTIONS OF BERNOULLI DISTRIBUTION
3. Success is denoted by :- p
Failure is denoted by :- q
sum of success & failure :- always unity
Thus, p + q =1
p =1-q and q=1-p

SYMMETRY OF BINOMIAL DISTRIBUTION
• Binomial distribution is symmetrical when p=0.5 or
when n is large.
• Binomial distribution is negatively skewed (peak
occurring to the right of the centre ) asymmetrical
distribution when p is greater than 0.5 ,p=0.9.
• Binomial distribution is positively skewed (peak
occurring to the left of the centre ) & asymmetrical
distribution when p is less than 0.5 ,p=0.1.

BINOMIAL DISTRIBUTION CRITERIA
There are two most important variables in the
binomial formula such as:
‘n’ it stands for the number of times the
experiment is conducted
‘p’ represents the possibility of one specific
outcome

MEAN AND VARIANCE OF A BINOMIAL DISTRIBUTION
Mean(µ) = np
Variance(σ2) = npq
S.D .( σ )= sqrt (npq)
The variance of a Binomial Variable is always less than
its mean. ∴ npq<np.

PROPERTIES OF BINOMIAL DISTRIBUTION
Properties of binomial distribution
Mean = N*P
example : throw coin 100 times N=100, P=.5,
thus mean is 100*.5 = 50
variance =N*p*q for coins : 100*.5*.5 =25
standard distribution = sqrt(25) = 5

BINOMIAL DISTRIBUTION – FORMULA
• First formula
• b(x,n,p)= nCx*Px*(1-P)n-x for x=0,1,2,…..n.
• where : –
b is the binomial probability.
x is the total number of successes.
p is chances of a success on an individual
experiment.
n is the number of trials

BINOMIAL DISTRIBUTION – FORMULA
• n>0 ∴ p,q≥0
• ∑b(x,n,p) = b(1) + b(2) + ….. + b(n) = 1
• Value of ‘n’ and ‘p’ must be known for applying
the above formula.
• So, we see that the existence of binomial
distribution highly depends on the knowledge of
these two parameters. This is why it is also
called bi-parametric distribution.

ALTERNATE FORMULA FOR BINOMIAL DISTRIBUTION
The first formula of binomial distribution can
calculate the possibility of success for the
binomial distribution. By using this formula you
can calculate it.
It seems easy but it’s not that easy to calculate
unless you are using a calculator. The calculator
can reduce a lot of your efforts and time.

• But if you want to do it manually then it might
take some time but you can solve it through
the following simple steps.
• If a coin is tossed 10 times then what is the
chances of getting exactly 6 heads?

 And if you are using the formula – b(x,n,p)= nCx*Px*(1-P)n-x
 The number of trials is 10 (n)
 The odds of success (tossing heads) is 0.5 (p)
 So, 1-p = 0.5, x= 6
 P(x=6) = 10C6 × 0.56 × 0.54 = 210 × 0.015625 × 0.0625 =
0.205078125
 With the help of the second formula, you can calculate the
binomial distribution.

PROBLEM
1. 80% of people those who purchase pet
insurance are women. If the owners of 9 pet
insurance are randomly selected, then find
the probability that exactly 6 out of them are
women.

SOLUTION
• Here are the steps.
• Find out the ‘n’ from the problem. Here n = 9
• Identify ‘X’. X = the number you are asked to search the
probability for is 6.
• (Divide the formula then it become easy to get the
solution) solve the first part of the formula: – n! / (n-X)!
X!
• Now add the variables = 9! (9-6)!*6! = 84. And keep it
aside for further uses.

SOLUTION
• Now find out the P and Q.
• P= the probable chances of success
and
• Q= the possibility of failure.

SOLUTION
 As mentioned in the above question p = 80% or 0.8
 so, the probability of failure = 1-0.8 = 0.2 (20%)
 Now let’s do the second part of the formula. Px = 0.86=
0.262144
 Q(n-x)= 0.2(9-6) = 0.23 = 0.008 (third part of the formula)
 Multiply the answer you get from step 3, 5, 6 together
 8×0.262144×0.008 = 0.176

SOLUTION
With the help of these two formulas, you can
calculate the binomial distributions easily.
The process to find out the binomial
calculation is not easy and is a little lengthy
process but the above-mentioned steps can
help in finding the solution.

Binomial distribution

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