Binomial Distribution – GCSE Statistics Edexcel Revision

Understanding Binomial Distribution

A binomial distribution is a specific type of discrete probability distribution.
It requires the probability experiment to be repeated a fixed number of times, with each trial being independent of the others.
It also states that there must be only two possible outcomes in a trial- success or failure- the probability of success remaining constant throughout.
Each event in a binomial experiment is independent of the others, which means the outcome of one event does not affect the outcome of any other event.

There are two parameters in a binomial distribution: the number of trials (n) and the probability of success in a single trial (p).
The mean of a binomial distribution is np, and the standard deviation is sqrt(npq), where q=1-p.
Binomial probabilities are calculated using the binomial formula: P(X=k) = nCk *pk * (1-p)n-k, where nCk represents the number of combinations of n trials taken k at a time.

In practical scenarios, binomial distribution is used when one is interested in the number of successes k in a fixed number of trials n for a binary process where the chance of success in each trial is the same, p.
It can also be used to conduct hypothesis testing and to build confidence intervals around sample estimates.

The outcomes of each trial are only of two categories, a success or a failure.
The probability of a success is the same in each trial.
The trials are independent, implying the outcome of one trial does not have any effect on the outcome of other trials.
The number of trials, n, is fixed in advance.

A binomial distribution can be a poor model if the probability of success is not the same in each trial or if the trials are not independent.
The spread or the variability of a binomial distribution increases as the probability p approaches 0.5. Hence, a binomial experiment with p near 0.5 will have more spread as compared to an experiment with p near 0 or 1.