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)
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
0.5 (probability of head/ tail in a single toss of a coin is ½), Cumulative = FALSE
The probability of getting exactly 6 heads =0.20507