Mean and Variance of Binomial Distribution (AS)

The binomial distribution is a probability distribution used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are labelled “success” and “failure”.
The random variable ‘X’ in a binomial distribution represents the number of successful outcomes in a given number of trials.
The mean (expected value) of a binomial distribution is given by the formula np, where n is the number of trials, and p is the probability of success on any individual trial.
The mean gives you the average number of successful outcomes you should expect if you conduct n trials.
The variance of a binomial distribution is given by the formula npq, where q = 1 - p (i.e. the probability of failure).
Variance measures how much, on average, each trial’s result will differ from the mean. In effect, it gives you a sense of how spread out your outcomes are likely to be around the average value.
If either the number of trials (n) or the success probability (p) increases whilst the other remains constant, the mean of the distribution will also increase.
Conversely, if either n or p decreases whilst the other remains constant, the mean will decrease.
An increase in the number of trials (n) whilst keeping p constant will lead to an increase in the variance. This is because, with more trials, there is more opportunity for the actual number of successes to deviate from the mean.
Contrarily, if the probability of success (p) increases towards 1 (with n remaining constant), the variance will decrease as nearly all trials will result in success, reducing the chance for deviation from the mean.
Memorizing these formulas and relationships will make calculating them in problems quicker, reducing time spent on these topics during revision. Remember to practise solving problems using these concepts to better understand their applications.
To ensure a deep understanding of binomial distribution, revise both its theoretical aspect, including derivation of formulae, and its practical applications in different scenarios.