Binomial Distribution
Statistics & Probability
Binomial Distribution: Formula, Examples & How to Calculate It
The binomial distribution tells you the probability of getting a fixed number of successes in a fixed number of independent yes-or-no trials. This guide breaks down the formula, the four conditions, the mean and variance, the normal approximation, and several fully worked examples. You’ll also see how the binomial distribution shows up in coin flips, exam pass rates, quality control, and clinical trials. By the end, you’ll be able to set up and solve a binomial probability problem from scratch.
Definition & Core Idea
What Is the Binomial Distribution?
The binomial distribution is a discrete probability distribution that gives the probability of getting exactly a certain number of successes out of a fixed number of independent trials, where each trial has only two possible outcomes. Think of flipping a coin ten times and asking, “what’s the probability I get exactly six heads?” That question, and thousands like it, is answered by the binomial distribution. Students run into this topic constantly in statistics coursework, and it’s also one of the building blocks behind hypothesis testing later in the semester.
The word “binomial” literally means “two names” or “two terms,” and that’s the entire spirit of the distribution: every single trial has exactly two outcomes. We usually label them success and failure, though those words are just labels. A “success” could be a defective product, a correct exam answer, a patient responding to treatment, or a voter choosing a candidate. Nothing about the word implies something good is happening.
2
Possible outcomes per trial — success or failure, with no third option allowed
4
Conditions that must all hold before you can call something a binomial setting
1700s
Era when Jacob Bernoulli’s work on repeated trials laid the foundation for this distribution
Where Did the Binomial Distribution Come From?
The mathematical roots of the binomial distribution trace back to the Swiss mathematician Jacob Bernoulli, whose work on probability and repeated independent trials was published posthumously in Ars Conjectandi in 1713. Bernoulli’s analysis of repeated yes-or-no experiments is why a single two-outcome trial is now called a Bernoulli trial, and why the binomial distribution is sometimes introduced as “the sum of n independent Bernoulli trials.” A century later, Abraham de Moivre showed that when the number of trials gets large, the shape of the binomial distribution starts to look like a smooth bell curve — an early version of what we now call the Central Limit Theorem.
You can read more about how this fits into the broader landscape of probability models in our guide to probability distributions, which covers how the binomial distribution relates to its discrete and continuous cousins.
Why Does the Binomial Distribution Matter for Students?
Almost every introductory statistics course in the United States and the United Kingdom devotes a full week to the binomial distribution because it’s the simplest model that captures a genuinely useful real-world pattern: repeated, independent, two-outcome events. According to the NIST Engineering Statistics Handbook, the binomial distribution is one of the most widely applied discrete distributions in quality control, reliability testing, and survey analysis. Once you understand it, distributions like the multinomial distribution and the Poisson distribution become much easier, because both are direct extensions of the same logic.
The simplest way to picture it: Imagine repeating the exact same yes-or-no experiment several times, where the outcome of one repetition has zero effect on the next. The binomial distribution tells you how likely each possible total count of “yeses” is.
When It Applies
The Four Conditions of a Binomial Experiment
Not every counting problem is a binomial distribution problem. Before you reach for the binomial formula, check that your scenario satisfies all four conditions below. Miss one, and you need a different model entirely — often the multinomial distribution, the hypergeometric distribution, or the Poisson distribution.
1. Fixed Number of Trials
The experiment is repeated a set number of times, denoted n, and that number is decided before data collection begins — not determined by the results themselves.
2. Only Two Outcomes
Each trial results in one of exactly two outcomes, usually called success and failure. There is no middle ground or third category.
3. Constant Probability
The probability of success, denoted p, stays the same from trial to trial. It cannot drift, increase, or decrease across the experiment.
4. Independence
The outcome of any one trial has no influence on the outcome of any other trial. Sampling with replacement typically guarantees this.
What Counts as a “Trial” in the Binomial Distribution?
A trial is simply one repetition of the underlying experiment. Flipping a coin once is a trial. Asking one randomly selected student whether they passed an exam is a trial. Inspecting one item off a production line is a trial. Each trial is a Bernoulli trial — named after Jacob Bernoulli — and the binomial distribution is built by stacking n of these trials together and counting how many came up “success.”
What Happens If Trials Aren’t Independent?
This is the condition students violate most often. If you draw cards from a deck without replacement, the probability of drawing a specific card changes after each draw, so the trials are no longer independent and p is no longer constant. In that situation, the correct model is the hypergeometric distribution, not the binomial distribution. However, when the population is large relative to the sample — say, surveying 50 people out of a population of 50,000 — the change in probability from one draw to the next is so small that the binomial distribution is still used as a very close approximation. This is a key idea covered in sampling distribution theory.
What If the Probability of Success Changes?
If p genuinely changes from trial to trial — for example, a basketball player’s free-throw percentage improving as a game goes on due to fatigue effects on the opposing team — the binomial distribution no longer applies cleanly. In practice, many real situations are treated as “approximately binomial” because the deviations are small enough not to matter for the purposes of the analysis.
Quick Self-Check Before You Start
Ask yourself: “Am I counting the number of times something happens out of a fixed number of attempts, where each attempt is the same and doesn’t affect the others?” If the answer is yes, you’re almost certainly looking at a binomial distribution problem.
Stuck on a Binomial Distribution Problem?
Our statistics tutors solve binomial probability, mean, variance, and normal approximation problems with full step-by-step working — delivered fast, 24/7.
Get Statistics Help Now Log InThe Core Formula
The Binomial Probability Formula Explained
Once the four conditions are met, the probability of getting exactly k successes in n trials, where each trial succeeds with probability p, is given by the binomial probability formula. This formula is the heart of the binomial distribution, and every worked example in this guide comes back to it.
Binomial Probability Mass Function
P(X = k) = C(n, k) · pk · (1 − p)n − k
What Does Each Symbol Mean?
- n — the fixed number of trials
- k — the number of successes you’re calculating the probability for (k can be 0, 1, 2, … up to n)
- p — the probability of success on a single trial
- 1 − p — the probability of failure on a single trial, often written as q
- C(n, k) — the binomial coefficient, also written as nCk or “n choose k,” representing the number of different ways k successes can be arranged among n trials
What Is the Binomial Coefficient and Why Is It There?
The binomial coefficient C(n, k) is calculated as n! / (k!(n − k)!), where the exclamation mark means factorial. This term exists because the formula needs to count every possible order in which k successes could occur among n trials. Getting heads on flips 1, 3, and 5 is a different sequence than getting heads on flips 2, 4, and 5, but both represent “3 heads out of 5 flips.” The binomial coefficient counts all of these equivalent sequences so they’re not missed or double-counted. This combinatorial logic connects directly to probability theory more broadly.
Worked Example — Coin Flips: Suppose you flip a fair coin 5 times. What’s the probability of getting exactly 3 heads? Here, n = 5, k = 3, and p = 0.5 (since the coin is fair).
First, C(5, 3) = 5! / (3! × 2!) = 10. Then, pk = (0.5)3 = 0.125, and (1 − p)n−k = (0.5)2 = 0.25.
Multiplying these together: 10 × 0.125 × 0.25 = 0.3125. So there’s about a 31.25% chance of getting exactly 3 heads in 5 flips.
What Does X Represent in P(X = k)?
X is called a random variable — it represents the number of successes that will occur, before the experiment is run. Once the experiment happens, X takes on one specific value. The binomial distribution is the full set of probabilities for every possible value X could take, from 0 successes all the way up to n successes. For a deeper dive into how random variables work in general, see our guide on discrete and continuous random variables.
Notation You’ll See in Textbooks
You’ll often see the binomial distribution written in shorthand as X ~ B(n, p) or X ~ Bin(n, p). This is read as “X follows a binomial distribution with parameters n and p.” Whenever you see this notation, it’s telling you exactly which formula to use and what numbers to plug in.
Summary Measures
Mean, Variance, and Standard Deviation of a Binomial Distribution
Beyond calculating individual probabilities, the binomial distribution has remarkably simple formulas for its center and spread. These come up constantly in expected value and variance problems, and they’re worth memorizing because they save enormous amounts of calculation time.
| Measure | Formula | What It Tells You | Plain-Language Meaning |
|---|---|---|---|
| Mean (Expected Value) | μ = n × p | The average number of successes you’d expect over many repetitions of the full experiment | If p = 0.3 and n = 100, expect about 30 successes on average |
| Variance | σ² = n × p × (1 − p) | How spread out the results tend to be around the mean | Larger when p is near 0.5; smaller when p is near 0 or 1 |
| Standard Deviation | σ = √(n × p × (1 − p)) | The typical distance between an observed outcome and the mean | Used to judge whether a result is “unusual” |
| Mode | Approximately floor((n + 1) × p) | The single most likely number of successes | The “peak” of the distribution’s shape |
Why Is the Mean Just n Times p?
This makes intuitive sense once you think about it from the bottom up. Each individual Bernoulli trial has an expected value equal to p (since it contributes 1 with probability p and 0 with probability 1 − p). The binomial random variable is just the sum of n such trials, and the expected value of a sum is the sum of the expected values. So n trials, each contributing p on average, gives a total expected value of n × p.
Worked Example — Mean and Standard Deviation
A multiple-choice quiz has 20 questions, and a student is guessing randomly on every question, with 4 options per question (so p = 0.25 for guessing correctly).
Mean: μ = n × p = 20 × 0.25 = 5. On average, a random guesser gets 5 questions right.
Variance: σ² = n × p × (1 − p) = 20 × 0.25 × 0.75 = 3.75.
Standard deviation: σ = √3.75 ≈ 1.94. So most random guessers will score somewhere around 5 ± 2 correct answers.
How Does the Shape of the Distribution Change with p?
When p = 0.5, the binomial distribution is perfectly symmetric. As p moves toward 0 or 1, the distribution becomes increasingly skewed. A small p (rare events, like manufacturing defects) produces a distribution heavily weighted toward 0 successes with a long tail toward higher counts — this is part of why the Poisson distribution is often used as an approximation in that situation. You can compare this visually against other shapes in our overview of distribution shape, skewness, and kurtosis.
Memory tip: Variance is largest when p = 0.5 because that’s when outcomes are “most uncertain” — both success and failure are equally likely on every trial, which maximizes the spread of possible totals.
Step-by-Step Process
How to Calculate a Binomial Probability: Step by Step
Solving a binomial distribution problem is a repeatable process. Once you’ve done it a handful of times, the steps become automatic. Here’s the process broken into six manageable stages.
1
Confirm the Four Conditions Apply
Re-read the problem and check for a fixed number of trials, two outcomes, constant probability, and independence. If any condition is missing, stop — you need a different distribution.
2
Identify n, k, and p
Write these three numbers down explicitly before doing any arithmetic. n is the total number of trials, k is the number of successes the question asks about, and p is the probability of success on one trial.
3
Calculate the Binomial Coefficient C(n, k)
Use C(n, k) = n! / (k!(n − k)!). For larger numbers, most calculators have an “nCr” button that does this directly.
4
Raise p to the Power of k
Calculate pk. This represents the probability of getting exactly k successes in a specific order.
5
Raise (1 − p) to the Power of (n − k)
Calculate (1 − p)n−k. This represents the probability of the remaining trials all being failures.
6
Multiply All Three Pieces Together
Multiply C(n, k), pk, and (1 − p)n−k. The result is P(X = k), the probability of exactly k successes.
Worked Example — Manufacturing Defects
A factory’s defect rate is 2%, meaning p = 0.02. In a random sample of 10 items, what’s the probability that exactly 1 item is defective?
n = 10, k = 1, p = 0.02. C(10, 1) = 10. p1 = 0.02. (1 − 0.02)9 = (0.98)9 ≈ 0.8337.
P(X = 1) = 10 × 0.02 × 0.8337 ≈ 0.1667, or about 16.7%.
Worked Example — “At Least” Probability
Using the same factory, what’s the probability that at least 1 item out of 10 is defective? “At least 1” is the complement of “exactly 0,” so it’s often faster to calculate the opposite case and subtract from 1.
P(X = 0) = C(10, 0) × (0.02)0 × (0.98)10 ≈ 1 × 1 × 0.8171 ≈ 0.8171.
P(X ≥ 1) = 1 − P(X = 0) = 1 − 0.8171 = 0.1829, or about 18.3%.
⚠️ Common trap: “At least” and “at most” problems are where students lose the most points. P(X ≥ k) is not the same as P(X = k), and P(X ≤ k) requires summing every probability from 0 up to k, not just calculating P(X = k) on its own.
Need a Full Binomial Distribution Assignment Solved?
From probability tables to mean, variance, and normal approximation problems — our statistics writers handle the entire assignment, matched to your course’s rubric and notation.
Start Your Order Log InBeyond a Single Outcome
Cumulative Binomial Probability: At Least, At Most, and Between
Most real assignment questions don’t ask for the probability of exactly k successes. They ask for ranges: “at least,” “at most,” “more than,” “fewer than,” or “between two values.” These all rely on the cumulative distribution function (CDF) of the binomial distribution, which adds up individual probabilities across a range of k values.
How Do You Translate Wording Into Math?
- “At least k” means P(X ≥ k) = P(X = k) + P(X = k+1) + … + P(X = n)
- “At most k” means P(X ≤ k) = P(X = 0) + P(X = 1) + … + P(X = k)
- “More than k” means P(X > k) = P(X ≥ k+1)
- “Fewer than k” means P(X < k) = P(X ≤ k−1)
- “Between a and b inclusive” means P(a ≤ X ≤ b) = P(X = a) + … + P(X = b)
For more background on how cumulative probabilities work across distributions in general, our guide to cumulative distribution functions walks through the underlying logic with both discrete and continuous examples.
Worked Example — “At Most” Problem
A call center has a 70% chance of resolving any given customer issue on the first call (p = 0.7). Out of 4 calls, what’s the probability that at most 2 are resolved on the first call?
n = 4, p = 0.7. We need P(X ≤ 2) = P(X=0) + P(X=1) + P(X=2).
P(X=0) = C(4,0)(0.7)0(0.3)4 = 1 × 1 × 0.0081 = 0.0081
P(X=1) = C(4,1)(0.7)1(0.3)3 = 4 × 0.7 × 0.027 = 0.0756
P(X=2) = C(4,2)(0.7)2(0.3)2 = 6 × 0.49 × 0.09 = 0.2646
P(X ≤ 2) = 0.0081 + 0.0756 + 0.2646 = 0.3483, or about 34.8%.
Why Do Textbooks Include Binomial Probability Tables?
Before calculators and software became standard, students relied on printed binomial probability tables that listed cumulative probabilities for common values of n and p. These tables are still included in many textbook appendices and exam formula sheets because they let you look up an answer instead of computing factorials by hand. They function similarly to the z-score table used for normal distribution problems — both exist to save time on repetitive calculations.
Related Distributions
Binomial Distribution vs. Normal and Poisson Distributions
One of the most frequently asked questions about the binomial distribution is how it relates to the distributions students encounter immediately afterward — the normal distribution and the Poisson distribution. All three describe probability, but each is built for a different kind of situation.
| Distribution | Type | Best Used When | Key Parameters |
|---|---|---|---|
| Binomial | Discrete | Fixed number of independent yes/no trials with constant probability of success | n (trials), p (probability of success) |
| Normal | Continuous | Large n with p not too close to 0 or 1 — approximates the binomial distribution’s shape | μ = np, σ = √(np(1−p)) |
| Poisson | Discrete | Large n, small p, counting rare events over a fixed interval — no fixed upper limit on count | λ = np |
| Multinomial | Discrete | Fixed number of trials, but more than two possible outcomes per trial | n (trials), p₁…pₖ (probabilities for each category) |
When Can You Use the Normal Approximation to the Binomial Distribution?
A common rule of thumb, cited across introductory statistics texts, is that the normal approximation works reasonably well when both np ≥ 10 and n(1−p) ≥ 10 (some textbooks use a threshold of 5). When these conditions hold, you can approximate a binomial distribution with a normal distribution that has mean μ = np and standard deviation σ = √(np(1−p)), then use z-scores to find probabilities instead of computing factorials directly.
What Is the Continuity Correction?
Because the binomial distribution is discrete (it only takes whole-number values) and the normal distribution is continuous (it takes every value along a line), a small adjustment called the continuity correction is applied. To approximate P(X = k) using the normal distribution, you actually calculate P(k − 0.5 < Y < k + 0.5), where Y is the normal approximation. This 0.5 adjustment accounts for the "gap" between discrete whole numbers.
Worked Example — Normal Approximation
A survey finds that 60% of college students (p = 0.6) prefer online textbooks. In a sample of 200 students (n = 200), estimate the probability that more than 130 prefer online textbooks.
Check conditions: np = 120, n(1−p) = 80. Both are well above 10, so the normal approximation is appropriate.
μ = np = 120. σ = √(200 × 0.6 × 0.4) = √48 ≈ 6.93.
With the continuity correction, P(X > 130) ≈ P(Y > 130.5). The z-score is (130.5 − 120) / 6.93 ≈ 1.51, which corresponds to roughly a 6.5% probability from a standard z-table.
When Should You Use the Poisson Approximation Instead?
The Poisson distribution becomes a better approximation than the normal distribution when n is large but p is very small — typically when np stays modest (often cited as np < 10) even as n grows large. This situation comes up constantly in rare-event modeling: the number of typos on a page, the number of equipment failures per month, or the number of patients in a hospital ward developing a rare side effect.
Real-World Use
Real-World Applications of the Binomial Distribution
The binomial distribution is not just a textbook exercise. It’s used by organizations across healthcare, education, government, and business in the United States and the United Kingdom every single day. Recognizing these patterns helps connect the math to something tangible.
Quality Control and Manufacturing
Manufacturers routinely sample a fixed number of items off a production line and count how many are defective. As long as the defect rate is roughly constant and items are sampled independently, the binomial distribution predicts how many defects to expect in a batch — and how unusual a higher-than-normal defect count would be. This directly supports decisions about whether to halt a production line for inspection.
Clinical Trials and Medicine
When researchers test whether a new drug is effective, they often track a binary outcome: did the patient respond to treatment or not? With a fixed number of participants and (under the trial’s design) a constant underlying probability of response, the binomial distribution underlies the statistical tests used to determine whether a drug’s effect is real or due to chance. This connects directly to hypothesis testing and the chi-square test for categorical outcomes.
Education and Exam Design
Multiple-choice exams are a textbook binomial scenario. If a student guesses randomly on every question, the number of correct answers follows a binomial distribution with p equal to 1 divided by the number of answer choices. Exam designers use this to set passing thresholds high enough that random guessing is unlikely to result in a passing score.
Marketing and Conversion Rates
Businesses track conversion rates — the percentage of website visitors who make a purchase, or the percentage of email recipients who click a link. If a company sends an email to a fixed list and each recipient independently decides to click or not, the number of clicks follows a binomial distribution. This lets marketing teams calculate the probability of hitting a campaign target.
Political Polling and Elections
When pollsters ask a fixed number of people whether they support a candidate, and treat each response as independent with a constant probability of support, the resulting count of “yes” responses follows a binomial distribution. This is the statistical foundation behind the margin of error reported alongside most political polls.
Genetics and Biology
In genetics, the inheritance of a trait that follows a simple dominant-recessive pattern can often be modeled with the binomial distribution. If each offspring independently has the same probability of inheriting a particular allele combination, the number of offspring with a given trait out of a litter follows a binomial distribution.
Spotting Binomial Situations in Assignment Prompts
Look for phrases like “out of n trials,” “what is the probability that exactly/at least/at most k…,” “each independently,” or “the same probability each time.” These phrases are strong signals that a binomial distribution approach is expected.
Errors to Avoid
Common Mistakes When Working With the Binomial Distribution
The binomial distribution looks simple once you’ve seen a few examples, but small errors creep in surprisingly often. Here’s what tends to go wrong, and how to fix it.
✓ Strong Practice
- Confirms all four binomial conditions before applying the formula
- Writes down n, k, and p explicitly before any calculation
- Treats “at least,” “at most,” and “exactly” as different problems requiring different sums
- Uses (1 − p) consistently for the failure probability
- Checks np ≥ 10 and n(1−p) ≥ 10 before applying a normal approximation
✗ Weak Practice
- Applies the binomial formula to situations without replacement or with changing probabilities
- Confuses k (number of successes) with n (number of trials)
- Calculates P(X = k) when the question asked for P(X ≥ k) or P(X ≤ k)
- Forgets the binomial coefficient C(n, k) entirely
- Uses the normal approximation when np or n(1−p) is too small
Mistake 1: Forgetting the Binomial Coefficient
It’s tempting to calculate pk(1−p)n−k and stop there. But this number alone only represents one specific ordering of successes and failures. Multiplying by C(n, k) accounts for every possible ordering that results in the same total count of successes.
Mistake 2: Mixing Up “Exactly,” “At Least,” and “At Most”
These three phrases describe completely different sums. “Exactly k” is a single term. “At least k” and “at most k” both require adding multiple terms together — or, more efficiently, subtracting from 1 when the complement is simpler to compute.
Mistake 3: Applying the Binomial Distribution to Dependent Events
Drawing cards without replacement, selecting items from a small finite population without replacement, or any scenario where the probability changes after each trial breaks the independence condition. In these cases, the hypergeometric distribution — not the binomial distribution — is correct, although the binomial is often used as a reasonable approximation when the population is large.
Mistake 4: Treating p as the Probability of the Event You Care About Without Checking the Wording
If a question describes a 92% pass rate but asks for the probability of failure, p should represent the probability of failure (0.08) for that specific question — or you flip which outcome you label “success” depending on what k represents. Always match p to whatever outcome k is counting.
⚠️ Double-check your final answer: A binomial probability must always be a number between 0 and 1. If your calculation produces a number outside that range, or a sum of all probabilities across k = 0 to n that doesn’t equal 1, go back and check your factorial calculations and exponents.
Frequently Asked Questions
Frequently Asked Questions About the Binomial Distribution
What is the binomial distribution in simple terms?
The binomial distribution describes the probability of getting a fixed number of successes in a set number of independent trials, where each trial has only two possible outcomes — success or failure — and the same probability of success every time. It answers questions like “what’s the chance of getting exactly 6 heads in 10 coin flips?”
What is the formula for the binomial distribution?
The formula is P(X = k) = C(n, k) × pk × (1 − p)n−k, where n is the number of trials, k is the number of successes, p is the probability of success on a single trial, and C(n, k) is the binomial coefficient representing the number of ways to arrange k successes among n trials.
What are the conditions for a binomial distribution?
Four conditions must hold: there is a fixed number of trials, each trial has exactly two possible outcomes, the probability of success stays constant across trials, and the trials are independent of one another. If any of these conditions fail, a different distribution is usually more appropriate.
How do you find the mean and variance of a binomial distribution?
The mean is calculated as μ = n × p, and the variance is calculated as σ² = n × p × (1 − p). The standard deviation is the square root of the variance. These formulas hold for every binomial distribution regardless of how large n is.
What is the difference between binomial and normal distribution?
The binomial distribution is discrete, meaning it only produces whole-number outcomes, while the normal distribution is continuous and can take any value along a range. When the number of trials is large and the probability of success is not too close to 0 or 1, the binomial distribution’s shape closely resembles a normal distribution, which is why the normal approximation to the binomial is commonly used.
What is the difference between binomial and Poisson distribution?
The binomial distribution requires a known, fixed number of trials and a constant probability of success on each one. The Poisson distribution instead models the number of times an event occurs within a fixed interval of time or space, without a fixed maximum number of trials. The Poisson distribution can be used as an approximation to the binomial distribution when n is large and p is small.
Can the binomial distribution be used for more than two outcomes?
No. The binomial distribution strictly applies to situations with exactly two outcomes per trial. When each trial can result in three or more categories — for example, rolling a die and recording which face comes up — the multinomial distribution is the correct generalization to use instead.
What is a Bernoulli trial and how does it relate to the binomial distribution?
A Bernoulli trial is a single experiment with exactly two possible outcomes, typically labeled success and failure, where the probability of success is constant. The binomial distribution describes the total count of successes when a fixed number of independent, identical Bernoulli trials are performed. In other words, the binomial distribution is built directly from repeated Bernoulli trials.
How is the binomial distribution used in real life?
The binomial distribution is used in quality control to count defective items in a sample batch, in clinical trials to model whether patients respond to treatment, in education to model exam guessing patterns, in marketing to model click-through and conversion rates, and in political polling to model the count of survey responses favoring a particular option.
What does it mean if a problem says “sampling without replacement” — can I still use the binomial distribution?
Sampling without replacement technically violates the independence and constant-probability conditions, since removing an item changes the composition of what remains. The exact model for this situation is the hypergeometric distribution. However, when the total population is much larger than the sample size, the binomial distribution is commonly used as a close and acceptable approximation.
How do I know whether to use a normal approximation for a binomial problem?
A widely used guideline is to check that both n × p and n × (1 − p) are at least 10 (some textbooks allow a threshold of 5). If both conditions are satisfied, the binomial distribution can be approximated by a normal distribution with mean n × p and standard deviation the square root of n × p × (1 − p), typically combined with a continuity correction of 0.5.