Binomial Distribution Calculator

How to Use the Binomial Distribution Calculator

This binomial distribution calculator finds the probability of obtaining a specified number of successes or a range of successes in a fixed number of trials. With the tool, you can calculate the probability of getting:

• exactly x successes, P(X = x)
• at most x successes, P(X ≤ x)
• less than x successes, P(X < x)
• at least x successes, P(X ≥ x)
• more than x successes, P(X > x)
• between a and b successes, P(a ≤ X ≤ b)

To use the calculator:

1. Enter the number of trials, n.
2. Enter the probability of success, p, on each trial as a decimal.
3. Enter the required number of successes, x. For a probability between two values, enter both the lower and upper values.
4. Select the probability type that matches your question.
5. Select how many decimal places you want in the final answer.
6. Click Calculate.

The calculator instantly returns the binomial probability based on the selected probability type, along with a simple interpretation, as well as the mean and standard deviation of the binomial distribution. You’ll also see a clear, step-by-step solution, showing you exactly how the selected probability was computed, step-by-step.

Which Probability Type Should I Choose?

The wording of a binomial problem tells you which probability to calculate. The table below provides a summary of the most common terms used in binomial probability problems, along with their correct probability in symbols.

Tip. Pay close attention to whether the endpoint is included. For example, at least 5 includes 5, whereas more than 5 does not.

What Is a Binomial Distribution?

A binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials.

For example, suppose you flip a fair coin 5 times and define getting heads as a success. The possible values of X are: 0, 1, 2, 3, 4, and 5. Since flipping a coin has two possible outcomes (success or failure), and the probability of success p = 0.5 (50-50 chance of getting a head), then the problem can be modelled using a binomial distribution.

Note. Although the outcome variable is called success, it does not always mean something positive. The success in a binomial distribution denotes the outcome of interest. For instance, if the goal is to count defective products, then a defective product is the outcome of interest.

When Should You Use the Binomial Distribution?

Before using the binomial formula, make sure your problem meets the conditions of a binomial experiment. These conditions help you determine whether a problem can be modelled using a binomial distribution or not.

Therefore, for a binomial experiment, the following conditions must be met:

1. There should be a fixed number of trials. The number of trials (n) is known before the experiment begins.
2. Two possible outcomes on each trial (success or failure).
3. The probability of success (p) must remain the same from one trial to another.
4. The outcome of one trial should be independent of another trial.

An example of a binomial experiment is flipping a fair coin 10 times. This satisfies the four conditions of a binomial experiment because there are:

• Fixed number of trials ( n= 10)
• Two possible outcomes (either get a head or a tail)
• Independent trials, since the result of one coin flip does not change or influence the result of any other flip.
• Constant probability ( p = 0.5) because the chance of getting heads stays exactly the same at 50% for every single flip.

Binomial Distribution Formula

By definition, the binomial probability formula is: P(X = x) = ⁿC_x × p^x × (1 − p)^{n − x}

Where:

• n is the number of trials
• x is the number of successes
• p is the probability of success on one trial
• 1 − p is the probability of failure
• ⁿC_x is the number of ways to choose x successes from n trials

The combination term is: ⁿC_x = n! / [x!(n − x)!].

The binomial probability formula helps us find the probability of getting exactly x successes in n trials. To find cumulative probabilities such as at most, at least, less than, more than, or between, we calculate the exact probabilities and sum them to get the required probability.

How to Calculate Binomial Probability

To calculate binomial probability manually, we apply the binomial probability formula. The binomial formula helps you calculate exact probabilities such as P(X=3). However, for a cumulative probability such as P(X ≤ 3), P(X ≥ 3), or P(2 ≤ X ≤ 4), we need to calculate several exact probabilities and sum them. Sometimes, we may need to use the complement rule to solve probabilities such as those asking for more than or at least.

To help you learn how to compute exact and cumulative binomial probabilities manually, follow these three examples.

Example 1: Exact Binomial Probability, P(X = x)

An AP Statistics student answers 8 independent multiple-choice questions. Based on past practice, the probability of answering any one question correctly is 0.65. What is the probability that the student answers exactly 5 questions correctly?

Solution

From the question, we know that:

• Number of trials, n = 8
• Probability of success, p = 0.65
• Number of successes, x = 5

We need to find the exact probability that the student gets 5 questions correct (i.e., P(X = 5)).

By definition, the binomial probability formula is: P(X = x) = ⁿC_x × p^x × (1 − p)^{n − x}

Substituting the values into the formula, we get:

P(X = 5) = ⁸C₅ × (0.65)⁵ × (1 − 0.65) ^{8− 5}

= 56 × (0.65)⁵ × (0.35) ³

= 0.2786

Therefore, P(X = 5) = 0.2786. This implies that the probability that the student answers exactly 5 of the 8 questions correctly is approximately 0.2786.

Example 2: Cumulative Binomial Probability (At Most)

A library at a Canadian college finds that 20% of borrowed laptops are returned late. Assume the return status of each laptop is independent. If 8 laptop loans are selected at random, what is the probability that at most 2 are returned late?

Solution

From the question, we know that:

• Number of trials, n = 8
• Probability of a late return, p = 0.20
• Number of successes, x = 2

In this case, the outcome variable is the late return.

“At most 2” means 0, 1, or 2 laptops are returned late. Thus, we need to find P(X ≤ 2)

Therefore, P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

This means we need to find the exact probabilities (P(X=0), P(X = 1), and P(X = 2)) and sum them.

Using the binomial distribution formula, we can calculate each of the probabilities as follows:

P(X = 0) = ⁸C₀ × (0.20)⁰ × (1 − 0.20) ^{8− 0}

= 1 × 1 × 0.16777216

= 0.16777

P(X = 1) = ⁸C₁ × (0.20)¹ × (1 − 0.20) ^{8− 1}

= 8 × 0.20¹ × 0.80 ⁷

= 0.33554

P(X = 2) = ⁸C₂ × (0.20)² × (1 − 0.20) ^{8− 2}

= 28 × 0.20² × 0.80 ⁶

= 0.29360

Since we have the exact probabilities, we can find the required probability as follows:

P(X ≤ 2) = 0.16777 + 0.33554 + 0.29360

= 0.79691

This implies that the probability that at most 2 of the 8 borrowed laptops are returned late is approximately 0.7969.

Example 3: Cumulative Binomial Probability (At Least)

A parcel delivery company in Australia finds that 30% of its deliveries arrive before noon. Assume that the delivery times are independent. If 6 deliveries are selected at random, what is the probability that at least 3 arrive before noon?

Solution

From the question, we know that:

• Number of trials, n = 6
• Probability of a delivery arriving before noon, p = 0.30
• Number of successes, x = 3

In this case, the outcome variable is a delivery arriving before noon.

“At least 3” means 3, 4, 5, or 6 deliveries arrive before noon. Thus, we need to find P(X ≥ 3).

We could calculate and add all four probabilities. However, it is easier to use the complement rule.

Using the complement rule, P(X ≥ 3) = 1 − P(X ≤ 2)

Thus, we only need to find P(X ≤ 2) and later subtract the result from 1.

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

Using the binomial probability formula, we can calculate the exact probabilities as follows:

P(X = 0) = ⁶C₀ × (0.30)⁰ × (1 − 0.30)^{6 − 0}

= 1 × 1 × 0.7⁶

Simplifying, we have:

= 1 × 1 × 0.11765

= 0.11765

P(X = 1) = ⁶C₁ × (0.30)¹ × (1 − 0.30)^{6 − 1}

= 6 × 0.30 × 0.7⁵

Simplifying, we get:

= 6 × 0.30 × 0.16807

= 0.30253

P(X = 2) = ⁶C₂ × (0.30)² × (1 − 0.30)^{6 − 2}

= 15 × 0.30² × 0.7⁴

Simplifying, we get:

= 15 × 0.09 × 0.2401

= 0.32414

Hence, P(X ≤ 2) = 0.11765 + 0.30253 + 0.32414

= 0.74432

Recall. P(X ≥ 3) = 1 − P(X ≤ 2)

Substituting P(X ≤ 2) into the formula: P(X ≥ 3) = 1 − 0.74432

= 0.25568

As such, the probability that at least 3 of the 6 deliveries arrive before noon is 0.2557.

Mean and Standard Deviation of a Binomial Distribution

Besides the requested probability, the calculator also reports the mean and standard deviation of the binomial distribution.

The mean of a binomial distribution, also known as its expected value, represents the average number of successes you would expect over many repetitions of the same binomial experiment. Its formula is: u = np

For instance, if the probability of success, p = 0.5, and the number of successes, n = 10, we would calculate the expected value as follows:

μ = 10 × 0.5

= 5

This implies that if the experiment were repeated many times, we would expect an average of 5 successes in 10 trials

The standard deviation of a binomial distribution describes how much the number of successes typically varies around the mean. Its formula is: σ = √[np(1 − p)]

Assuming n = 10 and p = 0.5, we would calculate the standard deviation as follows:

σ = √[10 × 0.5 × (1 − 0.5)]

= √2.5

= 1.5811

This means that, on average, the actual number of successes in any given set of 10 trials will typically vary by about 1.58 successes above or below that mean of 5.

Common Mistakes When Calculating Binomial Probability

Here are some common mistakes to avoid:

• Confusing “at least” with “more than.” At least 5 includes 5. More than 5 starts at 6.
• Confusing “at most” with “less than.” At most 5 includes 5. Less than 5 ends at 4.
• Using x as the number of failures. In the formula, x represents the number of outcomes you defined as successes.
• Using the binomial model when trials are not independent.

Normal Approximation to the Binomial Distribution

When the number of trials is large, calculating several binomial probabilities manually can become time-consuming. In such cases, the normal distribution can sometimes be used to approximate a binomial probability.

A common rule of thumb is that the normal approximation is appropriate when both np ≥ 5 and n(1 − p) ≥ 5, where n is the number of trials and p is the probability of success. Some textbooks use a stricter cutoff of 10 for both values.

Because the binomial distribution is discrete while the normal distribution is continuous, a continuity correction is usually applied when finding the probability.

If your problem asks you to approximate a binomial probability using the normal distribution, use our Normal Approximation to Binomial Calculator to calculate the probability and see the steps.

Frequently Asked Questions