If there are...
Number of events (n)
Probability of success per event (p)
%
What is the probability of getting...
exactly r successes
Number of successes (r)
Probability of r successes
%
Mean number of successes
Variance
Standard deviation

# Binomial Distribution Calculator

By Jakub Janus, PhD and Jasmine J Mah
Last updated: Mar 03, 2021

This binomial distribution calculator is here to help you with probability problems in the following form: what is the probability of a certain number of successes in a sequence of events? Read on to learn what exactly is the binomial probability distribution, when and how to apply it, and learn the binomial probability formula. Find out what is binomial distribution, and discover how binomial experiments are used in various settings.

## What is the binomial probability?

Imagine you're playing a game of dice. To win, you need exactly three out of five dice to show a result equal to or lower than 4. The remaining two dice need to show a higher number. What is the probability of you winning?

This is a sample problem that can be solved with our binomial probability calculator. You know the number of events (it is equal to the total number of dice, so five); you know the number of successes you need (precisely 3); you also can calculate the probability of one single success occurring (4 out of 6, so 0.667). This is all the data required to find the binomial probability of you winning the game of dice.

Note that to use the binomial distribution calculator effectively, the events you analyze must be independent. It means that all the trials in your example are supposed to be mutually exclusive.

The first trial's success doesn't affect the probability of success or the probability of failure in subsequent events, and they stay precisely the same. In the case of a dice game, these conditions are met: each time you roll a die constitutes an independent event.

Sometimes you may be interested in the number of trials you need to achieve a particular outcome. For instance, you may wonder how many rolls of a die are necessary before you throw a six three times. Such questions may be addressed using a related statistical tool called the negative binomial distribution. Make sure to read about the differences between this distribution and the negative binomial distribution.

## Binomial probability formula

To find this probability, you need to use the following equation:

`P(X=r) = nCr * pʳ * (1-p)ⁿ⁻ʳ`

where:

• n is the total number of events;
• r is the number of required successes;
• p is the probability of one success;
• nCr is the number of combinations (so-called "n choose r"); and
• P(X=r) is the probability of an exact number of successes happening.

You should note that the result is the probability of an exact number of successes. For example, in our game of dice, we needed precisely three successes - no less, no more. What would happen if we changed the rules so that you need at least three successes? Well, you would have to calculate the probability of exactly three, precisely four, and precisely five successes and sum all of these values together.

You can use the SMp(x) probability distribution to simulate many other distributions including the binomial one. Make sure to give it a try!

## How to use the binomial distribution calculator: an example

Let's solve the problem of the game of dice together.

1. Determine the number of events. n is equal to 5, as we roll five dice.
2. Determine the required number of successes. r is equal to 3, as we need exactly three successes to win the game.
3. The probability of rolling 1, 2, 3, or 4 on a six-sided die is 4 out of 6, or 0.667. Therefore p is equal to 0.667 or 66.7%.
4. Calculate the number of combinations (5 choose 3). You can use the combination calculator to do it. This number, in our case, is equal to 10.
5. Substitute all these values into the binomial probability formula above:

`P(X = 3) = 10 * 0.6673 * (1-0.667)(5-3) = 10 * 0.6673 * (1-0.667)(5-3) = 10 * 0.296 * 0.333 * 2 = 2.96 * 0.111 = 0.329`

1. You can also save yourself some time and use the binomial distribution calculator instead :)

## How to calculate cumulative probabilities

Sometimes, instead of an exact number of successes, you want to know the probability of getting r or more successes or r or less successes. To calculate the probability of getting any range of successes:

1. Use the binomial probability formula to calculate the probability of success (P) for all possible values of r you are interested in.
2. Sum the values of P for all r within the range of interest.

For example, the probability of getting two or fewer successes when flipping a coin four times (p = 0.5 and n = 4) would be:

`P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)`

`P(X ≤ 2) = 37.5% + 25% + 6.25%`

`P(X ≤ 2) = 68.75%`

This calculation is made easy using the options available on the binomial distribution calculator. You can change the settings to calculate the probability of getting:

• Exactly r successes: P(X = r)
• r or more successes: P(X ≥ r)
• r or fewer successes: P(X ≤ r)
• Between r₀ and r₁ successes P(r₀ ≤ X ≤ r₁)

## Binomial probability distribution experiments

The binomial distribution turns out to be very practical in experimental settings. However, the output of such a random experiment needs to be binary: pass or failure, present or absent, compliance or refusal. It's impossible to use this design when there are three possible outcomes.

At the same time, apart from rolling dice or tossing a coin, it may be employed in somehow less clear cases. Here are a couple of questions you can answer with the binomial probability distribution:

• Will a new drug work on a randomly selected patient?
• Will a light bulb you just bought work properly, or will it be broken?
• What is a chance of correctly answering a test question you just drew?
• What is a probability of a random voter to vote for a candidate in an election?
• How likely is it for a group of students to be accepted to a prestigious college?

Experiments with precisely two possible outcomes, such as the ones above, are typical binomial distribution examples, often called the Bernoulli trials.

In practice, you can often find the binomial probability examples in fields like quality control, where this method is used to test the efficiency of production processes. The inspection process based on the binomial distribution is designed to perform a sufficient number of checkups and minimize the chances of manufacturing a defective product.

If you don't know the probability of an independent event in your experiment (`p`), collect the past data in one of your binomial distribution examples, and divide the number of successes (`y`) by the overall number of events `p = y/n`. You can convert a percentage to a fraction, or the other way around, by using this calculator.

Once you have determined your rate of success (or failure) in a single event, you need to decide what's your acceptable number of successes (or failures) in the long run. For example, one defective product in a batch of fifty is not a tragedy, but you wouldn't like to have every second product faulty, would you?

Bernoulli trials are also perfect at solving network systems. Interestingly, they may be used to work out paths between two nodes on a diagram. This is the case of the Wheatstone bridge network, a representation of a circuit built for electrical resistance measurement.

Like the binomial distribution table, our calculator produces results that help you assess the chances that you will meet your target. However, if you like, you may take a look at this binomial distribution table. It tells you what is the binomial distribution value for a given probability and number of successes.

## Mean and variance of binomial distribution

One of the most exciting features of binomial distributions is that they represent the sum of a number `n` of independent events. Each of them (`Z`) may assume the values of 0 or 1 over a given period.

Let's say the probability that each `Z` occurs is `p`. Since the events are not correlated, we can use random variables' addition properties to calculate the mean (expected value) of the binomial distribution `μ = np`.

The variance of a binomial distribution is given as: `σ² = np(1-p)`. The larger the variance, the greater the fluctuation of a random variable from its mean. A small variance indicates that the results we get are spread out over a narrower range of values.

The standard deviation of binomial distribution, another measure of a probability distribution dispersion, is simply the square root of the variance, `σ`. Keep in mind that the standard deviation calculated from your sample (the observations you actually gather) may differ from the entire population's standard deviation. If you find this distinction confusing, there here's a great explanation of this distinction.

There's a clear-cut intuition behind these formulas. Suppose this time that I flip a coin 20 times:

• My `p` is then equal to 0.5 (unless, of course, the coin is rigged);
• Each `Z` has an equivalent chance of 0 or 1;
• The number of trials, `n`, is 20. This sequence of events fulfills the prerequisites of a binomial distribution.

The mean value of this simple experiment is: `np = 20 * 0.5 = 10`. We can say that on average if we repeat the experiment many times, we should expect heads to appear ten times.

The variance of this binomial distribution is equal to `np(1-p) = 20 * 0.5 * (1-0.5) = 5`. Take the square root of the variance, and you get the standard deviation of the binomial distribution, `2.24`. Accordingly, the typical results of such an experiment will deviate from its mean value by around 2. Hence, in most of the trials, we expect to get anywhere from 8 to 12 successes.

Use our binomial probability calculator to get the mean, variance, and standard deviation of binomial distribution based on the number of events you provided and the probability of one success.

## Other considerations

Developed by a Swiss mathematician Jacob Bernoulli, the binomial distribution is a more general formulation of the Poisson distribution. In the latter, we simply assume that the number of events (trials) is enormous, but the probability of a single success is small.

The binomial distribution is closely related to the binomial theorem, which proves to be useful for computing permutations and combinations. Make sure to check out our permutations calculator, too!

Keep in mind that the binomial distribution formula describes a discrete distribution. The possible outcomes of all the trials must be distinct and non-overlapping. What's more, the two outcomes of an event must be complementary: for a given `p`, there's always an event of `q = 1-p`.

If there's a chance of getting a result between the two, such as 0.5, the binomial distribution formula should not be used. The same goes for the outcomes that are non-binary, e.g., an effect in your experiment may be classified as low, moderate, or high.

However, for a sufficiently large number of trials, the binomial distribution formula may be approximated by the Gaussian (normal) distribution specification, with a given mean and variance. That allows us to perform the so-called continuity correction, and account for non-integer arguments in the probability function.

Maybe you still need some practice with the binomial probability distribution examples?

Try to solve the dice game's problem again, but this time you need three or more successes to win it. How about the chances of getting exactly 4?

Jakub Janus, PhD and Jasmine J Mah