Binomial Distribution
The term “binomial distribution”, in statistics, refers to a type of discrete distribution which reflects the probability of getting one of two mutually exclusive results in a set of identical experiments under certain circumstances.
The results might be presented as a success vs a failure, a yes or no dichotomy, 1 or 0 in the binary system, or any other form that suggest only two options available. An important notion is that only one result is possible in each experiment, so the results must be contradicting each other. Examples of situations with such results might be found everywhere, from testing the effectiveness of a drug, or forecasting the victory of a certain candidate, to simple instances like tossing a coin. So, binomial distribution is applied in many spheres to determine the likelihood of a certain result being received in a given number of times when conducting a given number of experiments.
Binomial Distribution main features
As it has been stated above, a binomial distribution is a discrete one, which means that the results presented in the distribution are counted separately. A binomial distribution reflects the occurrence of only two possible results, and by this fact it is opposed to a normal distribution, which estimates a probability of multiple possible outcomes.
A set of experiments with only two outcomes available is also required to meet certain criteria for its results to be gauged with the use of a binomial distribution.
These necessary conditions are the following:
- There must be a fixed number of experiments or trials, which cannot be altered, or otherwise the binomial distribution would be different.
- Trials or experiments of the observed set must be conducted with no dependence or relation to each other, in other words, an outcome of one trial has no effect on another trial.
- There is a fixed probability of success in each case of trial or experiment, which isn’t changed regardless the number of trials.
A noteworthy feature of the binomial distribution is that one of its instances is a set of Bernoulli trials. A Bernoulli trial means an experiment with two possible results (namely a failure and a success), which is very close to the definition of experiments examined with the use of the binomial distribution. So, a binomial distribution for only one experiment is a Bernoulli distribution.
Binomial Distribution formula
When it’s necessary to find out what’s the possibility of a certain result in a row of trials (for example, how many times a coin would come up tails if it is tossed 10 times), a binomial distribution is calculated. To perform such a calculation, it’s required to know the number of experiments or trials (the number is usually marked as “n” in the formula) and the possibility of a result considered to be a success (marked as “p”). These figures are then used to calculate the mean, or expected value, which is done by multiplying (n * p).
The number of results considered to be successes is also necessary, and it’s marked as “x” in the formula. One more crucial figure in the formula is the combination C of n and x (it’s important to note that a factorial is used to find it out).
The full formula for calculating the binomial distribution is the following:
