Binomial Probability Calculator

Use the Binomial Calculator to compute individual and cumulative binomial probabilities. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems.

To learn more about the binomial distribution, go to Stat Trek's lesson on the binomial distribution.

Select a probability from the dropdown box.
Enter a value in each unshaded textbox.
Click Calculate to compute probabilities.

Probability
Probability of success on a trial
Number of trials
Number of successes (x)
Minimum successes (x₁)
Maximum successes (x₂)
P(X=x)
P(X<x)
P(X≤x)
P(X>x)
P(X≥x)
P(x₁ ≤ X ≤ x₂)
P(X < x₁) + P(X > x₂)
Mean (μ)
Standard deviation (σ)

Frequently-Asked Questions

Calculator | Sample Problem

Instructions: To find the answer to a frequently-asked question, simply click on the question.

What is a binomial experiment?

A binomial experiment has the following characteristics:

The experiment involves repeated trials.
Each trial has only two possible outcomes - a success or a failure.
The probability that any trial will result in success is constant.
All of the trials in the experiment are independent.

A series of coin tosses is a perfect example of a binomial experiment. Suppose we toss a coin three times. Each coin flip represents a trial, so this experiment would have 3 trials. Each coin flip also has only two possible outcomes - a Head or a Tail. We could call a Head a success; and a Tail, a failure. The probability of a success on any given coin flip would be constant (i.e., 50%). And finally, the outcome on any coin flip is not affected by previous or succeeding coin flips; so the trials in the experiment are independent.

What is a binomial distribution?

A binomial distribution is a probability distribution . It refers to the probabilities associated with the number of successes in a binomial experiment.

For example, suppose we toss a coin three times and suppose we define Heads as a success. This binomial experiment has four possible outcomes: 0 Heads, 1 Head, 2 Heads, or 3 Heads. The probabilities associated with each possible outcome are an example of a binomial distribution, as shown below.

Outcome, x	Binomial probability, P(X = x)	Cumulative probability, P(X < x)
0 Heads	0.125	0.125
1 Head	0.375	0.500
2 Heads	0.375	0.875
3 Heads	0.125	1.000

How do you compute binomial probability?

This calculator uses the binomial formula to compute binomial probability:

Binomial Formula. Suppose a binomial experiment consists of n trials and the probability of success on an individual trial is p. Then the binomial probability that the experiment results in x successes is:

P(X = x)

\[ \binom{n}{x} \]

* p^x * (1 - p)^{n - x}

or
P(X = x) = _nC_x * p^x * (1 - p)^{n - x}
or
P(X = x) = { n! / [ x! (n - x)! ] } * p^x * (1 - p)^{n - x}

Note: Each of the binomial formulas shown above are equivalent. If you take the AP Statistics exam, the first formula is the one that is provided to you at the exam. So, if you take the exam, it would be prudent to know how to compute binomial probability from that formula.

What is the number of trials?

The number of trials refers to the number of replications in a binomial experiment.

Suppose that we conduct the following binomial experiment. We flip a coin and count the number of Heads. We classify Heads as success; tails, as failure. If we flip the coin 3 times, then 3 is the number of trials. If we flip it 20 times, then 20 is the number of trials.

Note: Each trial results in a success or a failure. So the number of trials in a binomial experiment is equal to the number of successes plus the number of failures.

What is the number of successes?

Each trial in a binomial experiment can have one of two outcomes. The experimenter classifies one outcome as a success; and the other, as a failure. The number of successes in a binomial experient is the number of trials that result in an outcome classified as a success.

What is the probability of success on a single trial?

In a binomial experiment, the probability of success on any individual trial is constant. For example, the probability of getting Heads on a single coin flip is always 0.50. If "getting Heads" is defined as success, the probability of success on a single trial would be 0.50.

What is the binomial probability?

A binomial probability refers to the probability of getting EXACTLY r successes in a specific number of trials. For instance, we might ask: What is the probability of getting EXACTLY 2 Heads in 3 coin tosses. That probability (0.375) would be an example of a binomial probability.

In a binomial experiment, the probability that the experiment results in exactly x successes is indicated by the following notation: P(X=x);

What is the cumulative binomial probability?

Cumulative binomial probability refers to the probability that the value of a binomial random variable falls within a specified range.

The probability of getting AT MOST 2 Heads in 3 coin tosses is an example of a cumulative probability. It is equal to the probability of getting 0 heads (0.125) plus the probability of getting 1 head (0.375) plus the probability of getting 2 heads (0.375). Thus, the cumulative probability of getting AT MOST 2 Heads in 3 coin tosses is equal to 0.875.

Notation associated with cumulative binomial probability is best explained through illustration. The probability of getting FEWER THAN 2 successes is indicated by P(X<2); the probability of getting AT MOST 2 successes is indicated by P(X≤2); the probability of getting AT LEAST 2 successes is indicated by P(X≥2); the probability of getting MORE THAN 2 successes is indicated by P(X>2).

What is the mean and standard deviation of a binomial distribution?

The mean (μ) and standard deviation (σ) of a binomial distribution are calculated using the number of trials (n) and the probability of success (P) on each trial.

The mean is the average number of successes you'd expect over many repetitions of the experiment.

μ = n * P

And the standard deviation measures the typical distance of the number of successes from the mean.

σ = sqrt [ n * P * (1 - P) ]

Sample Problem

Calculator | Frequently-Asked Questions

Suppose you toss a fair coin 12 times. What is the probability of getting exactly 7 Heads.

Solution:

We know the following:
- The probability of success (i.e., getting a Head) on any single trial is 0.5.
- The number of trials is 12.
- The number of successes is 7 (since we define getting a Head as success).
We select P(X = x) from the Probability dropdown box. Then, we plug the probability of success, the number of trials, and the number of successes into the Binomial Calculator; and we hit the Calculate button.

The calculator reports that the binomial probability is 0.193. That is the probability of getting EXACTLY 7 Heads in 12 coin tosses. (The calculator also reports the cumulative probabilities. For example, the probability of getting AT MOST 7 heads in 12 coin tosses is a cumulative probability equal to 0.806.)

Suppose the probability that a college freshman will graduate is 0.6. Three college freshmen are randomly selected. What is the probability that at most two of these students will graduate?

Solution:

We know the following:
- The probability of success for any individual student is 0.6.
- The number of trials is 3 (because we have 3 students).
- The number of successes is 2.
We select P(X = x) from the Probability dropdown box. With that selection, the calculator computes a number of probabilities, including the probability we are interested in - P(X ≤ x). Then, we plug the probability of success, the number of trials, and the number of successes into the Binomial Calculator; and we hit the Calculate button.

The calculator reports that the probability that two or fewer of these three students will graduate is 0.784.

Over his career as a professional golfer, Jack Nichlaus shot a birdie on 22% of the holes that he played. In a 72-hole match, what is the probability that Jack would shoot at least 14 birdies but no more than 17 birdies?

Solution:

We know the following:
- The probability of success (shooting a birdie) on any given hole is 0.22.
- The number of trials is 72.
- The minimum number of successes is 14.
- The maximum number of successes is 17.
Since we want to compute a cumulative probability, we select P(x₁ ≤ X ≤ x₂) from the Probability dropdown box. Then, we plug the probability of success, the number of trials, the minimum number of successes, and the maximum number of successes into the Binomial Calculator ; and we hit the Calculate button.

The calculator reports that the probability that Jack Nichlaus would shoot between 14 and 17 birdies in a 72-hole match is 0.43107.