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A binomial experiment is any statistical experiment with the following properties;

  • Has n repeated trials
  • Each trial has just two possible outcomes; success and a failure.
  • The probability of success is constant in every trial
  • The trials are independent of each other, i.e the outcome of one trial does not affect the outcomes of the other.

Below are Binomial Experiment Examples;

  • Example 1: Flipping a coin
  • Example 2: Results of an exam (pass or fail)
  • Example 3: Gender of individual (male or female)
  • Example 4: 10 year survival diagnosis of a disease (alive or dead)

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Example 1: Flipping a coin

Flipping a coin for a given n number of times is an binomial experiment since there are n trials each with just two outcomes in every trial(head or tail).The probability of head(success) remains the same throughout the experiment(0.5).The outcome of any trial does not influence the outcome in the other.

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Explanation of the Example

A random variable is said to follow a binomial distribution if it has n repeated trials and has only two outcomes in each trial. Binomial probability is the probability associated with a binomial experiment in exactly x number of successes. For instance, if a coin is flipped twice and we are interested in the number of tails (successes), we are likely to get either no tail, 1 tail or two tails as shown below;

No. of tails Probability
0 0.25
1 0.5
2 0.25

A binomial distribution has 3 main properties

The mean = n * p

Variance = n * p (1-p)

Standard deviation = √ (n*p (1-p))

Where n is the total number of trials, p is the probability of success and 1-p is the probability of failure.

The Binomial distribution formula is therefore given by;

binomial distribution formula

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Binomial Distribution Real Life Applications

Basically, real life applications of binomial distribution is anything that you can think of whose outcomes are two: success or failure. Some of the Binomial Experiment Examples are presented below;

  • If a new drug is introduced to cure a given disease. It is usually administered to n number of subjects and the results are that it will either cure the disease (success) or it doesn’t cure the disease (failure).
  • In betting, when you place bet, you are either likely to win(success) or lose(failure)
  • If you purchase a lottery ticket, you are either going to win the money (success) or aren’t (failure).
  • In Casinos, you are likely to either draw the correct card or he wrong card.

Derivation of Poisson distribution from binomial probability distribution function

Poisson distribution is derived from the binomial probability distribution function. Check here for the entire process

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