Notes on Binomial Distribution | Grade 12 > Mathematics > Probability | KULLABS.COM

Binomial Distribution

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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:

  • 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 )



  • 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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