Statistics - Binomial Distribution

Bionominal appropriation is a discrete likelihood conveyance. This distribution was discovered by a Swiss Mathematician James Bernoulli. It is used in such situation where an experiment results in two possibilities - success and failure. Binomial distribution is a discrete probability distribution which expresses the probability of one set of two alternatives-successes (p) and failure (q). Binomial distribution is defined and given by the following probability function −

Formula

${P(X-x)} = ^{n}{C_x}{Q^{n-x}}.{p^x}$

Where −

${p}$ = Probability of success.
${q}$ = Probability of failure = ${1-p}$.
${n}$ = Number of trials.
${P(X-x)}$ = Probability of x successes in n trials.

Example

Problem Statement −

Eight coins are tossed at the same time. Discover the likelihood of getting no less than 6 heads.

Solution −

Let ${p}$=probability of getting a head. ${q}$=probability 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 \times \frac{1}{256} + 8 \times \frac{1}{256} + 1 \times \frac{1}{256}, \\[7pt] \, = \frac{37}{256}$

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