Binomial Distribution

Subject: Mathematics

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Overview

This note gives you the knowledge on Binomial distribution theorem when it is used and also that mean and variance of the distribution
Binomial Distribution

Random Variable and Probability Distributions

A variable that takes different probability in a random way so that the outcomes are not known is called a random variable. Hence, X is called a random variable if it takes values X­1 ­, X­2 , ­X­3­. . .X­n randomly. There are several types of probability distribution such as the binomial distribution, Poisson, Normal etc. They differ according to the nature of the random variable.

Bernoulli Process:

An experiment consisting only two outcomes is known as “Bernoulli Process”. Generally, the two outcomes are called success and failure. For example, tossing a coin, result of an examination, throwing a dice, production of bulbs etc.

Binomial Distribution:

The discrete probability distribution derived from the Bernoulli process is known as Binomial distribution. The following are the basic assumptions under which Binomial Distribution can be used:

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  • Random experiment should be performed for a fixed number of times.
  • The experiment should only have two events known as success or failure.
  • All the experiments performed should be independent of one another.
  • The probability of success denoted by p should be constant per every experiment.

When the probability of a success in one trial is known, the probabilities of success of exactly once, twice, thrice, . . . etc. in n trials can also be known.

Let a trial be repeated so as to make a set of n trials. We denote the occurrence of an event known as success by S and the non-occurrence, a failure by F. Let p and q be the probabilities of a success and a failure in one trial respectively be such that p + q = 1. Let us assume that the trials are independent and the probability of success in every trial is the same. Now, we find the probabilities of 0, 1, 2 , . . . .,n success in n trials. Thr probabilities of a success and a failure be denoted by P(S) and P(F) respectively. Then the probability of r success and (n-r) failure in a set of n trials in any specified order say S.S.S.S …S (r times) F.F.F.F.F.F…F.(n-r times).

=P(S).P(S)… P(S) P(F).P(F).P(F)….. p(F)

=(p.p.p……r times) (q.q.q.q…. (n-r)times)

= pr . qn-r

But the number of orders of occurring r success out of n trials is same as the number of combination of n things taken r at a time i.e. n­r­. These are all equally probable and mutually exclusive hence by the theorem of total probability of r success; i.e. P(r) in a set of n independent trials is given by:

$$\;P(r)\;$$=^nC_r\;p^r\;q^_{n-r}\;\;$$……\;(0\leq\;r\leq\;n)\;$$

Thus if n = number of trials performed

p = probability of success in a trial

q = probability of a failure in a trial such that p + q = 1

r = number of success \;in trials

then P(r) = P(x = r) = probability of r success in n trials

$$\;=\;^nC_r\;p^r\;q^_{n-r}\;$$

The probabilities of 0, 1, 2, 3, . . . . . . n successes obtained by putting r = 0, 1, 2, 3, . . . . n in the above equation is listed below:

No of successes (r)

Probability of r success i.e. P(r)

0

P(0) = qn

1

P(1) = C(n,1)p1qn-1

2

P(2) = C(n,2) P2qn-2

3

P(3) = C(n,3) P3qn-3

n

P(2) = C(n,n) Pnqn-n = pn

Mean and Standard deviation of Binomial Distribution:

If p be the probability of a success and q that of a failure in one trial, then the probabilities of 0,1,2,3, . . . . n successes in n trials are listed above which are the successive terms of the binomial expansion of (p+q)n . Hence the distribution is known as binomial distribution. The mean and the standard deviation of the binomial distribution and np and (npq)-1/2 respectively.the two independent constants n and p (or q) are known as parameters.

  • Mean of the distribution is given by np
  • Variance of the distribution is given by npq
  • $$Standard\;deviation\;of\;the\;distribution\;is\;given\;by\;\sqrt{npq}\;$$

    Taken reference from

    ( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )

Things to remember
  • P(r) = P(x = r) = probability of r success in n trials

    $$\;=^nC_r\;p^r\;q^_(n-r)\;$$

  • Mean of the distribution is given by np

  • Variance of the distribution is given by npq

  • $$Standard\;deviation\;of\;the\;distribution\;is\given\;by\;\sqrt{npq}\;$$

 

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