/    /  Statistics – Binomial Distribution

Binomial Distribution:

A Binomial distribution is a discrete probability distribution in which the random variable (X) follows:

  • When there are only two possible outcomes of each trial, success and failure.
  • Here, probability (success) is p and the probability (failure) is q or (1-p) where either of them remains constant throughout experiment.
  • Experiment consisting of ‘n’ finite number of trials.
  • Each trial is independent of the last.
  • Outcomes are mutually exclusive and the sum of their probabilities is complementary (p+ q= 1).

Binomial probability formula is given by

P(x) =  ∙ px ∙ (1-p)n-x , where  =

The mean of  binomial distribution=  E[X] = E[X1+X2+X3+….Xn] = p + p + p+…..+p = np

n times

Example:

we can calculate the probability that two of the next three babies born are male using binomial distribution.

The variance of binomial distribution is Var[X] = np(1-p)

Mode = Binomial Distribution 5 (i2tutorials.com)

The Binomial distribution with a single trial (n = 1), is Bernoulli distribution.

Syntax for calculating binomial distribution using Excel is BINOMDIST(number_s,trials,probability_s,cumulative)

where ,Number_s is the number of successes

Trials is the number of independent trials

Probability_s is the probability of success in a single trial

Cumulative is a logical value which determines the form of the function. If TRUE, then BINOMDIST returns the cumulative distribution function (at most n successes), if FALSE, it returns the probability mass function (exactly n successes).

Example:

If a coin is tossed 10 times. The probability of getting exactly 6 heads is ?

Here, number_s = 6 , Trails = 10, Probability

s =
Binomial Distribution 4 (i2tutorials.com)

0.5 (probability of head/ tail in a single toss of a coin is ½), Cumulative = FALSE

Binomial Distribution 2(i2tutorials.com)

The probability of getting exactly 6 heads =0.20507

Binomial Distribution 3 (i2tutorials.com)