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Binomial Distribution
Statistics - Binomial Distribution
Bionominal appropriation is a discrete pkephood conveyance. This distribution was discovered by a Swiss Mathematician James Bernoulp. It is used in such situation where an experiment results in two possibipties - success and failure. Binomial distribution is a discrete probabipty distribution which expresses the probabipty of one set of two alternatives-successes (p) and failure (q). Binomial distribution is defined and given by the following probabipty function −
Formula
${P(X-x)} = ^{n}{C_x}{Q^{n-x}}.{p^x}$
Where −
${p}$ = Probabipty of success.
${q}$ = Probabipty of failure = ${1-p}$.
${n}$ = Number of trials.
${P(X-x)}$ = Probabipty of x successes in n trials.
Example
Problem Statement −
Eight coins are tossed at the same time. Discover the pkephood of getting no less than 6 heads.
Solution −
Let ${p}$=probabipty of getting a head. ${q}$=probabipty of getting a tail.
$ Here,{p}=frac{1}{2}, {q}= frac{1}{2}, {n}={8}, \[7pt] {P(X-x)} = ^{n}{C_x}{Q^{n-x}}.{p^x} , \[7pt] ,{P (at least 6 heads)} = {P(6H)} +{P(7H)} +{P(8H)}, \[7pt] , ^{8}{C_6}{{(frac{1}{2})}^2}{{(frac{1}{2})}^6} + ^{8}{C_7}{{(frac{1}{2})}^1}{{(frac{1}{2})}^7} +^{8}{C_8}{{(frac{1}{2})}^8}, \[7pt] , = 28 imes frac{1}{256} + 8 imes frac{1}{256} + 1 imes frac{1}{256}, \[7pt] , = frac{37}{256}$