Binomial Distribution Notes (Year 12) – A-Level Maths

The binomial distribution is a special type of discrete probability distribution. It is used to model situations where there are a fixed number of trials, each with two possible outcomes, and the probability of success is constant.

Coin Toss Example

Consider tossing a biased coin 4 times, where:

P(head) = 0.3, P(tail) = 0.7

Let X be the number of heads obtained, so X is a discrete random variable. We are interested in its probability distribution:

P(X = x) for x = 0, 1, 2, 3, 4

Conditions for Using a Binomial Model

A random variable can be modelled using a binomial distribution if all the following conditions are satisfied:

1. Fixed number of trials (n)

• In our example, n = 4 tosses.

2. Independent trials

• Each coin toss does not affect the others.

3. Two possible outcomes per trial

• Here, “head” is considered a success and “tail” a failure.

4. Constant probability of success (p)

• The probability of getting a head is the same in each trial: p = 0.3.

Example Distribution Table

This table shows all possible values of X (number of heads) and their associated outcomes for 4 coin tosses with P(head) = 0.3.

Probability Mass Function (PMF) – Binomial Distribution

If X is binomially distributed, write:

X ∼ B(n, p)

Where:

• n = number of trials

• p = probability of success

The probability of obtaining exactly r successes is:

Example Calculation

In the coin toss example, you can calculate P(X = 2) as:

Mean and Variance

For a binomial distribution:

Example Calculation

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:

pearson-edexcel-a-level-gce-in-mathematics-formulae-book.pdf

Download PDF • 2.35MB

Summary

The binomial distribution is ideal for counting successes in a fixed number of independent trials with two outcomes and a constant probability of success.Remember to:

1. Check the four conditions.

2. Use the PMF formula to find exact probabilities.

3. Calculate mean and variance using np and np(1−p).

4. Carefully interpret cumulative probability questions.