Binomial distribution formula: definition, uses and examples

Knowing how to use binomial distribution is one of the most important elements in understanding statistics, and with a range of real-world applications, it’s certainly something your students will need to have a good handle on.

It’s important for them to know both the principles behind this and how to put them into practice using their calculators.

Casio fx-CG100 graphic calculator in the classroom

Key features of binomial distribution

In order to consider using the binomial distribution formula, a question must meet some essential criteria. These are:

1. A fixed number of observations or trials
2. Each trial must be independent
3. There are exactly two potential outcomes of each trial (success and failure)
4. The probability of success is exactly the same in each trial

One of the simplest examples of this that students should be able to grasp easily is coin tosses. In this example, we can ensure a fixed number of trials by specifying how many times the coin is tossed. As each toss is independent and the probability of it landing heads or tails does not change, it fulfils all the criteria for the binomial distribution formula.

This can then be used to calculate the probability of getting a certain number of heads for a given number of tosses.

The binomial distribution formula

A-level maths students will need to understand the binomial distribution formula so they can work out the probability of obtaining a specific number of successes across a fixed number of independent trials. While calculators can simplify these operations, recognising what each part of the formula represents is an important part of developing confidence with binomial probability questions.

Binomial probability formula

Where:
● n = the total number of trials
● r = the number of successes
● p = the probability of success on a single trial
● $C_r^n$ = the number of possible combinations, calculated as $C_r^n=\frac{n!}{r!\left(n-r\right)!}$

Alongside the probability formula, students studying binomial distribution in A-level statistics should also be familiar with mean and variance, as these are commonly used to interpret the expected outcome and spread of results.
Mean = np
Variance = np(1 − p)
Standard deviation = √(np(1 − p))

These measures describe the centre and spread of the discrete probability distribution. Your students should recognise that the mean represents the expected number of successes, while the variance and standard deviation of the binomial distribution indicate how much the results are likely to vary around that expected value.

How to evaluate single and cumulative binomial probabilities on a calculator

While it’s important that your students understand the binomial distribution formula, a scientific calculator can help them approach binomial probability questions more efficiently and check their working. The fx-991CW+ UK includes a dedicated Distribution app, allowing students to calculate both single and cumulative probabilities in just a few steps.

Calculating a single binomial probability (Binomial PD)

Suppose your students are asked to calculate the probability of getting exactly four successes in ten trials, where the probability of success on each trial is 1/3.

On the fx-991CW+ UK:
Ensure the calculator is reset if required.
From the HOME screen, select Distribution.
Choose Binomial PD.
Select Variable to calculate a single probability.
Enter the following values, pressing EXE after each one:
x = 4
N = 10
p = 1 ÷ 3
Press EXE again to calculate the result.

The calculator returns a probability of approximately 0.2276, meaning there is a 22.76% chance of obtaining exactly four successes.

Calculating cumulative probabilities (Binomial CD)

For cumulative probability questions, the process is almost identical. For example, your students may be asked to find the probability of obtaining four or fewer successes from the same binomial distribution.

To calculate this:

Open the Distribution app.
Select Binomial CD.
Choose Variable.
Enter:
x = 4
N = 10
p = 1 ÷ 3
Press EXE to display the cumulative probability.

This returns the probability that X ≤ 4. For this example, the calculator displays a probability of approximately 0.787, meaning there is a 78.7% chance of obtaining four or fewer successes in ten trials when the probability of success in each trial is 1/3. This makes Binomial CD particularly useful for answering exam questions that ask for the probability of achieving at most, no more than or up to and including a specified number of successes.

Classroom engagement with the Casio ClassWiz

Calculating several probabilities at once

If your students need to compare probabilities for several different values of x, they can select List instead of Variable.

After choosing List, enter each required value of x before entering N and p. When EXE is selected, the calculator displays the probability for every value entered, making it easy to compare outcomes or complete tables without repeating the same calculation multiple times.

As students become more confident with binomial distributions, they can also use their calculator to check the values they’ve calculated for the mean and standard deviation. This helps reinforce the relationship between probability calculations and the distribution’s key summary measures, while giving students greater confidence in their answers.

Exploring binomial distributions with the fx-CG100

If your students are using the fx-CG100 graphic calculator, they can build on these calculations using the Distribution app. Alongside calculating Binomial PD and Binomial CD values, students can visualise the distribution itself, making it easier to explore how changing the values of n or p affects its shape. This provides a useful way to compare different scenarios and helps develop a stronger understanding of how binomial distributions behave.

Applications of binomial distribution

Demonstrating how binomial distribution can be applied to real-world situations can help students get to grips with the formula and understand its uses. Some common applications for this include:
Finance: Identifying the probability of stocks or prices rising or falling based on past data, or understanding the rate of fraudulent transactions.
Retail: Calculating what percentage of sales are likely to be returned.
Manufacturing: Improving quality control by working out the likelihood of finding a certain number of faulty items.
Healthcare: Working out the probability of patients encountering side effects of a medication.

These can all be used to identify if real-world performance is better or worse than would be statistically expected and the impact of any changes to processes.

Common mistakes when applying the binomial distribution formula

There are a few common errors students might make when working out binomial distributions that teachers should look out for. These include misinterpreting the problem statement, or failing to identify the correct values for the number of trials, the probability of success or the desired outcome.

Using the right technology – such as a powerful calculator – can help eliminate these mistakes and ensure students are able to approach these questions with confidence when it comes to exams.

Whether you’re introducing binomial distributions or exploring another area of the maths curriculum, the Casio Education Resource Centre offers a wide range of teaching resources designed to help you get the most from Casio calculators in the classroom. You can find lesson ideas, guides and classroom activities to support your teaching across a variety of topics.

FAQs

Binomial distribution FAQs

What’s the formula for binomial distribution?

The binomial distribution formula calculates the probability of achieving a specific number of successes in a fixed number of independent trials. It’s written as P(X = x) = ⁿCₓ × pˣ × (1 − p)ⁿ⁻ˣ, where n is the number of trials, x is the number of successes and p is the probability of success in each trial.

How do you find the mean and standard deviation of a binomial distribution?

For a binomial distribution, the mean is calculated using np, while the standard deviation is √(np(1 − p)). Students studying binomial distribution in A-level statistics should understand these formulas alongside the probability formula, as they describe the expected value and spread of the distribution.

What’s the difference between Binomial PD and Binomial CD?

Binomial PD calculates the probability of obtaining exactly a specified number of successes. Binomial CD calculates the cumulative probability, returning the probability of obtaining that number of successes or fewer. Both functions are available in the Distribution app on compatible Casio calculators.

Can Casio calculators calculate binomial probabilities directly?

Yes. The fx-991CW+ UK and fx-CG100 both include a Distribution app that allows students to calculate Binomial PD and Binomial CD values quickly. This makes it easy to check manual calculations and solve binomial probability questions more efficiently.

When should students use the binomial distribution formula?

Students should use the binomial distribution formula when a probability question involves a fixed number of independent trials, only two possible outcomes (success or failure), and a constant probability of success. These are the key conditions for a binomial distribution.

What’s the difference between a binomial distribution and a normal distribution?

A binomial distribution is a discrete probability distribution, meaning it models the probability of a count of successes. A normal distribution is continuous and is used to model values that can take any value within a range. For larger values of n, a binomial distribution can sometimes be approximated using a normal distribution.

August 2026