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# Binomial Distribution notes (Year 12)

Updated: Mar 22, 2023

The binomial distribution is a special type of discrete probability distribution, used to model a wide variety of situations.

Coin Tossing Example

I toss a biased coin 4 times. Suppose that P(head) = 0.3 and P(tail) = 0.7.

Let X be the number of heads obtained, so X is a discrete random variable. We are interested in the distribution of X i.e. what is P(X=x) for x=0, 1, 2, 3, 4.

Conditions for a Binomial Distribution Model

• Fixed number (n) of trials ... here n=4

• Trials must be independent ... here the coin tosses are indeed independent

• Each trial has only two outcomes (success/failure) ... here we have head or tail

• Probability of success (p) is constant for each trial ... here p=0.3 is constant

Probability Mass Function

If X is binomially distributed, write X~B(n, p) where X is the number of successes in the n trials and p is the probability of success for each trial.

The probabilities can be calculated using this formula:

Mean and Variance

The mean (or expected value) of the binomial distribution is given by np and the variance by np(1-p).

Cumulative Probabilities

When questions are set in context there are different forms of words that can be used to ask for probabilities. The correct interpretation of these phrases is critical, especially when dealing with cumulative probabilities. The table below gives some examples.

The formula book contains Binomial Cumulative Distribution tables for certain values of n and p. Download the tables here.