Binomial Distribution: Formula and Examples

A binomial distribution is the probability of winning a certain number of times if you do something a fixed number of times. Say you flip a coin 10 times and want exactly 6 heads. That is a binomial probability question.

It's a discrete probability distribution, which just means it counts whole numbers. You can get 2 heads or 3 heads, but never 2.5.

The Four Rules Your Situation Must Follow

A binomial distribution only works when all four of these are true:

Fixed number of trials. You pick the number of tries (called n) before you start. Ten flips. Twenty items checked.
Two outcomes. Each try is a success or a failure. "Success" just means the thing you're counting, even if it's bad news, like a broken item.
Same chance every time. The success probability (called p) is constant.
Independent trials. One result doesn't change the next. A coin doesn't remember its last flip.

If even one rule fails, the formula will give you the wrong answer. A single try with two outcomes is called a Bernoulli trial. The binomial model simply counts successes across many of them.

The Binomial Distribution Formula

Here is the binomial formula for the chance of exactly k successes:

P(X = k) = C(n, k) × p to the power of k × (1 minus p) to the power of (n minus k)

Symbol	What it means
n	Number of trials
k	Number of successes you want
p	Chance of success on one try
1 minus p	Chance of failure on one try
C(n, k)	The number of ways to choose which tries are the successes

Where C(n, k) is the binomial coefficient. It is n! divided by k! times n minus k!. Without it, you'd only get the chance of one exact order, like heads first and tails last. Sometimes this entire equation is referred to as the probability mass function.

Solutions Examples

Example 1 6 heads in 10 flips

Flip a coin 10 times. What's the chance of exactly 6 heads? Here n is 10, k is 6, and p is 0.5.

There are 210 ways to pick which 6 flips are heads. Each full set of 10 flips has a 1 in 1,024 chance. So that's 210 out of 1024 or around.205. That is about 20%.

Example 2: At most 1 defect in 20 items

A machine makes 5% bad items. You check 20. What's the chance of 0 or 1 bad ones?

Zero bad: about 0.3585
Exactly one bad: about 0.3774
Together: about 0.736, or 74%

This "or fewer" idea is called cumulative probability. So the chance of 2 or more bad items is about 26%.

Example 3: At least one click from 30 emails

Each email has a 3% chance of a click. You send 30. What's the chance of at least one click?

Don't add up 30 answers. Do it backwards instead. The chance of zero clicks is about 0.401. So at least one click is 1 minus 0.401, which is about 0.599. That's close to 60%.

Mean, Variance, and Standard Deviation

The mean is what you'd expect on average. Just do n × p. Flip a coin 10 times and you'd expect about 5 heads.

Variance shows how much results jump around. It's n × p × (1 minus p). The standard deviation is the square root of that.

For the 20 items with a 5% defect rate, the mean is 1. The variance is 0.95. The standard deviation is about 0.97. So you'll usually see 0, 1, or 2 bad items. Seeing 10 would be shocking.

The graph changes with p. At 0.5 it's balanced, like a hill. With a small p, results bunch up near zero. With a big p, it's the mirror image.

Binomial vs Other Distributions

Not sure if the binomial distribution is the right pick? Use this table.

Your situation	Better choice
Only one try	Bernoulli distribution
Fixed tries, same chance, independent	Binomial
Small group, no putting items back	Hypergeometric distribution
Counting tries until the 3rd success	Negative binomial
Counting events over time, no set limit	Poisson distribution
The chance changes each try	Poisson binomial
Very many trials	Normal approximation

The normal approximation works well when n × p and n × (1 minus p) are both about 10 or more. Some books say 5, so follow your class.

Stats papers can be hard to read. AI research paper summaries can help you get the main points fast.

Common Mistakes

Skipping the four rules.
Thinking p never changes.
Mixing up "exactly" and "at most."
Forgetting the binomial coefficient.
Using it when there's no fixed number of tries.

Here's an example of the first one. Draw cards from a small deck and don't put them back, and the draws aren't independent. For the last one, think of phone calls in an hour. There's no set n, so Poisson fits better.

Check Your Work with Tools

A binomial calculator can catch small slips. Try one of these:

Excel or Google Sheets: =BINOM.DIST(6, 10, 0.5, FALSE) for exactly 6. Use TRUE for "at most 6."
Python: binom.pmf(6, 10, 0.5) for exactly 6, binom.cdf(6, 10, 0.5) for at most 6. Import it first with from scipy.stats import binom.
R: dbinom(6, 10, 0.5) and pbinom(6, 10, 0.5)

Want to see how researchers use this? You can find research papers on the topic.

FAQ

What is a binomial distribution?

It counts successes in a fixed number of attempts. The success of each try is independent of the previous one.

What is the difference between binomial and normal?

Binomial counts whole numbers. Normal covers every value on a smooth curve. With many tries, binomial starts to look like a bell curve.

What is the difference between binomial distribution and Poisson distribution?

The number of trials in a binomial is fixed. Poisson counts events with no set limit, like emails in an hour.

How do I find the chance of "at most k" successes?

Add the chances from 0 up to k. In Excel, pick TRUE for the cumulative option.

What is Canyam?

Canyam is a website where you can search research papers and read AI generated summaries of them.

Final Thoughts

Check the four rules first. If your problem passes, use the formula, then double check with a tool. Try it on something you really count, like clicks or defects. Once the binomial distribution feels easy, you'll find more guides on the Canyam blog.