A binomial distribution table is a statistical reference chart that lists the probability of obtaining exactly k successes in n independent trials, given a fixed probability of success p per trial. Rows index the number of successes k; columns index the success probability p. It gives you P(X = k) directly, eliminating the need to compute C(n,k) · pk · (1−p)n−k by hand.
Binomial Probability Calculator
Click any cell to highlight the probability. Values shown to 4 decimal places. Values < 0.0001 shown as 0.0000.
Binomial Distribution Table PDF — Free Download
Download a free printable binomial probability table in PDF format. Covers n=1–20 and the most common p values at four decimal places — ready for exams and coursework.
What Is a Binomial Distribution?
The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success p.
4 Conditions (FINS)
Key Parameters
Real-World Examples
- 10 coin flips: how many heads?
- Quality control: defects in a batch of 20 items
- A/B testing: conversions in 50 ad impressions
- Drug trial: patients who respond out of 15
- Multiple-choice guessing: correct answers in 5 questions
What Is the Binomial Probability Formula?
The binomial probability formula is P(X=k) = C(n,k) × pk × (1−p)n−k, where C(n,k) = n! / (k!(n−k)!) is the binomial coefficient counting the number of ways to arrange k successes in n trials.
Mean (Expected Value)
Average number of successes. For n=10, p=0.3: μ = 3.0
Variance
For n=10, p=0.3: σ² = 10 × 0.3 × 0.7 = 2.1
Standard Deviation
For n=10, p=0.3: σ = √2.1 ≈ 1.449
How to Read a Binomial Distribution Table (Step-by-Step)
Reading a binomial distribution table takes five steps: identify n, find the p column, locate the k row, read the intersection, and sum rows for cumulative probabilities.
What Is the Difference Between Exact and Cumulative Binomial Tables?
An exact binomial table gives P(X = k) — the probability of getting exactly k successes. A cumulative binomial table gives P(X ≤ k) — the probability of getting k or fewer successes, found by summing P(X=0) through P(X=k).
Exact Table — P(X = k)
Shows the probability of exactly k successes. Used when the question asks "what is the probability of getting exactly 3 heads in 10 flips?" Switch to the Exact tab above.
Cumulative Table — P(X ≤ k)
Shows the probability of k or fewer successes. Preferred for hypothesis testing, confidence intervals, and "at most" problems. Switch to the Cumulative tab above.
Worked Examples Using the Binomial Distribution Table
Example 1 — Finding an Exact Probability
Question: A fair coin is flipped 8 times. What is the probability of getting exactly 3 heads?
Example 2 — Finding P(X ≤ k) — Cumulative
Question: A basketball player makes 60% of free throws. She shoots 6 attempts. What is the probability she makes fewer than 4?
Example 3 — Finding P(X ≥ k) — Complement Rule
Question: A quality control test has a 15% defect rate. In a batch of 10 items, what is the probability that at least 2 are defective?
How to Use Binomial Distribution on a TI-84 Calculator
The TI-84 has two built-in binomial functions: binompdf(n, p, k) for exact probability P(X=k), and binomcdf(n, p, k) for cumulative probability P(X≤k).
binompdf — Exact P(X = k)
binompdf(n, p, k)binomcdf — Cumulative P(X ≤ k)
binomcdf(n, p, k)Common Binomial Probabilities: Quick Reference
These are the most frequently looked-up exact probabilities in coursework and exams. All use p = 0.50 (fair coin/equal probability).
How Does the Binomial Distribution Compare to Other Distributions?
The binomial distribution is best when n is fixed and trials are independent. Use the Poisson distribution for rare events without a fixed n; use the normal approximation when np ≥ 5 and n(1−p) ≥ 5.
| Distribution | When to Use | Key Parameter | Trials Fixed? |
|---|---|---|---|
| Binomial | Fixed n, two outcomes, constant p | n, p | ✅ Yes |
| Poisson | Rare events, rate λ, no fixed n | λ (rate) | ❌ No |
| Geometric | Trials until first success | p | ❌ No |
| Normal (approx) | Large n (np≥5, n(1−p)≥5) | μ=np, σ=√npq | ✅ Yes |
| Negative Binomial | Trials until rth success | r, p | ❌ No |
Binomial Distribution: Key Statistics & Facts
Sources & Further Reading
The binomial distribution is foundational in probability theory. These peer-reviewed and institutional resources provide rigorous treatment of the topic:
The authoritative U.S. government reference for the binomial distribution PMF, CDF, mean, and variance. Includes worked engineering applications. This page's formulas are consistent with the NIST standard.
The leading graduate-level textbook on mathematical statistics. Chapter 5 provides full derivation of the binomial PMF, moment generating function, and the relationship to Bernoulli trials. The mean (μ=np) and variance (σ²=npq) formulas on this page follow DeGroot & Schervish's notation.
Harvard's introductory probability course uses the binomial distribution as its first major discrete distribution (Lecture 3). The FINS conditions on this page derive from Blitzstein's characterization of binomial experiments. Freely available via OpenCourseWare.
Standard undergraduate engineering statistics textbook. The quality control and acceptance sampling examples on this page align with Walpole et al.'s treatment of industrial binomial applications, including the normal approximation rule np≥5.
1 Bernoulli, J. (1713). Ars Conjectandi. Basel. The binomial theorem and its application to probability was formally published posthumously in this work.
Frequently Asked Questions About the Binomial Distribution Table
What is a binomial distribution table?
A binomial distribution table is a statistical reference chart that lists the probability of obtaining exactly k successes in n independent trials, given a fixed probability of success p per trial. Rows represent k values; columns represent p values. It eliminates the need to calculate P(X=k) = C(n,k) · pk · (1−p)n−k manually.
How do you read a binomial distribution table?
Find the block for your n. Locate the column for your p. Move to the row for k (your desired number of successes). The cell value is P(X = k). For cumulative probability P(X ≤ k), sum rows from k=0 down to your target k. For P(X ≥ k), use the complement: 1 − P(X ≤ k−1).
What is the mean and variance of a binomial distribution?
For a binomial distribution B(n, p): mean = μ = np; variance = σ² = np(1−p); standard deviation = σ = √(np(1−p)). For example, B(10, 0.4): μ = 4, σ² = 2.4, σ ≈ 1.549. These formulas are far simpler than the general formulas and are derived directly from the structure of Bernoulli trials.
When should you use the binomial distribution?
Use the binomial distribution when all four FINS conditions are met: (1) Fixed number of trials n; (2) Independent trials; (3) N outcomes — exactly two per trial (success/failure); (4) Same probability p on every trial. If n is very large and p is small, the Poisson distribution is a better fit. If np≥5 and n(1−p)≥5, the normal approximation works well.
What is the difference between binomPDF and binomCDF?
binomPDF (probability density/mass function) gives P(X = k) — the probability of exactly k successes. binomCDF (cumulative distribution function) gives P(X ≤ k) — the total probability of k or fewer successes. On a TI-84: binompdf(n, p, k) returns the exact value; binomcdf(n, p, k) returns the cumulative. The tables on this page show both.
How do you find P(X ≥ k) from the table?
Use the complement rule: P(X ≥ k) = 1 − P(X ≤ k−1). Look up P(X ≤ k−1) in the cumulative table, then subtract from 1. Example: P(X ≥ 4) with n=10, p=0.3. Find P(X ≤ 3) from the table = 0.6496. Then P(X ≥ 4) = 1 − 0.6496 = 0.3504.
What is a Bernoulli trial and how does it relate to the binomial distribution?
A Bernoulli trial is a single random experiment with exactly two possible outcomes — success (probability p) or failure (probability 1−p). The binomial distribution is the sum of n independent, identical Bernoulli trials. If X ~ Bernoulli(p), then X₁ + X₂ + ... + Xₙ ~ Binomial(n, p). This is why n=1 in the binomial distribution gives the Bernoulli distribution as a special case.
Can I use the binomial table for p > 0.5?
Yes. This table includes p values from 0.05 to 0.95. For p > 0.5, many textbooks use the symmetry property: P(X = k | n, p) = P(X = n−k | n, 1−p). So P(3 successes in 10 trials at p=0.7) equals P(7 successes in 10 trials at p=0.3). The calculator on this page handles all p values directly.
What are real-world applications of the binomial distribution?
The binomial distribution is used across many fields: quality control (defect rates in manufacturing), clinical trials (patient response rates), polling and survey analysis (proportion supporting a policy), A/B testing in digital marketing (conversion rates), sports analytics (shot success rates), and finance (probability of n out of N assets gaining in a period). Anywhere fixed-trial, binary-outcome experiments occur, the binomial table applies.
Where can I download a free binomial distribution table PDF?
Free PDF downloads are available at the top of this page. The standard PDF covers exact probabilities P(X=k) for n=1–20 and p=0.05–0.95 at four decimal places. The cumulative PDF shows P(X≤k) for the same range. Both are formatted for exam printing and include the binomial formula at the top.
Understanding the Binomial Distribution Table
Why the Table Has p Values from 0.05 to 0.95
Tables typically cover p=0.05 to 0.95 because probabilities outside this range produce values so close to 0 or 1 that manual calculation is trivial. For p > 0.5, use the symmetry property: P(X=k | n, p) = P(X=n−k | n, 1−p) to use the lower half of the table.
Why Probabilities Sum to 1.0000 Across Each Row Block
For any fixed n and p, the values in the k column from k=0 to k=n are exhaustive and mutually exclusive outcomes. Their sum must equal 1. Small rounding to 4 decimal places may cause a sum of 0.9999 or 1.0001 — this is normal and not an error in the table.
Normal Approximation: When the Table Isn't Enough
For n > 20, binomial tables become unwieldy. When np ≥ 5 and n(1−p) ≥ 5, use the normal approximation: X ≈ N(np, np(1−p)). Apply a continuity correction: P(X = k) ≈ P(k−0.5 < Z < k+0.5) using the Z-table.