Binomial Distribution Table
Explore exact and cumulative binomial probabilities in an interactive table. Select the number of trials and probability of success, then highlight a value to see what it means and how it is calculated.
Binomial Probability Distribution Table
Choose the number of trials (n), the probability of success (p), and the probability view. Each row represents a possible number of successes (k), while each column represents a success probability. The selected cell shows the probability for your chosen combination.
Supported inputs: integer n from 0 to 10,000, p from 0 to 1, and integer k from 0 to n. Custom p appears as a separate column when it is not a preset. For large n, change k to jump to its row window.
For n = 10 and p = 0.5, the chance of three or fewer successes is approximately 17.19%.
≈ marks rounded values. Very small probabilities use scientific notation; numeric limits do not mean mathematical impossibility or certainty. CDF values always include the full lower tail, not just visible rows.
All possible success counts shown.
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How to Read a Binomial Distribution Table
A binomial table answers probability questions about a fixed number of independent trials. First, find the number of trials n. Next, locate the probability p and the number of successes k. Finally, check whether the table displays exact probabilities or cumulative probabilities before reading the selected value.
Binomial PMF vs. Cumulative Binomial Table
The probability mass function (PMF) gives P(X = k), the chance of exactly k successes.
The cumulative distribution function (CDF) gives P(X ≤ k), the chance of k or fewer successes.
For at least k successes, use P(X ≥ k) = 1 − P(X ≤ k − 1). For more than k successes, use P(X > k) = 1 − P(X ≤ k).
Do not interchange these expressions: including or excluding the boundary changes the answer.
These are mathematical identities. The lookup sums the requested events directly, preserving small upper tails instead of subtracting a rounded CDF from 1.
Binomial Distribution Formula
For X following a binomial distribution with n trials and success probability p:
P(X = k) = C(n,k) p^k (1 − p)^(n−k)
Here, C(n,k) is the number of ways to choose k successes from n trials. The model assumes independent trials, two possible outcomes per trial, and a constant success probability.
P(X ≤ k) = Σᵢ₌₀ᵏ C(n,i)pⁱ(1 − p)ⁿ⁻ⁱ
Mean = np. Variance = np(1 − p). Standard deviation = √[np(1 − p)].
Worked Examples Using the Binomial Table
These examples use sample values, not your current inputs.
Example 1 — Exactly three successes
Suppose n = 10 and p = 0.5. The probability of exactly three successes is:
P(X = 3) = C(10,3)(0.5)^3(0.5)^7 = 0.1171875.
The probability is approximately 11.72%.
Example 2 — At most three successes
For the same distribution:
P(X ≤ 3) = 0.171875.
This includes zero, one, two, and three successes. The probability is approximately 17.19%.
Example 3 — At least four successes
P(X ≥ 4) = 1 − P(X ≤ 3) = 0.828125.
The probability of four or more successes is approximately 82.81%.
When to Use a Binomial Distribution Table
A binomial model is appropriate when the experiment has a fixed number of independent trials, each trial has two possible outcomes, and the probability of success remains constant.
If success probabilities vary between trials, events are dependent, or sampling changes the probabilities substantially, a standard binomial model may not be appropriate.
For event counts over an interval rather than a fixed number of trials, compare the Poisson Distribution Table.
Frequently Asked Questions
What is a binomial distribution table?
A binomial distribution table shows the probabilities associated with different numbers of successes in a fixed number of independent trials. Depending on the view, it displays exact probabilities P(X = k) or cumulative probabilities P(X ≤ k).
How do I read a binomial distribution table?
Select the number of trials n, find the success probability p, and locate the row for the number of successes k. Read the value where that row and column intersect. Check whether you are using the PMF or CDF view.
What is the difference between PMF and CDF in a binomial table?
PMF gives the probability of exactly k successes. CDF gives the probability of k or fewer successes, including k itself.
How do I find the probability of at least k successes?
Use P(X ≥ k) = 1 − P(X ≤ k − 1). For at least four successes, subtract the cumulative probability through three successes from 1.
Can I use a binomial table for any number of trials?
The mathematical distribution is defined for nonnegative integer n, but printed tables usually cover selected trial counts. This interactive table supports custom values within its stated input limits and displays a manageable range of rows for large distributions.
What does p mean in a binomial distribution?
The parameter p is the probability of success on each individual trial. It must be between 0 and 1 and must remain constant across the trials.
Why does my table show 0.0000 for a probability?
A displayed value of 0.0000 may be a very small positive probability rounded to four decimal places. This table uses scientific notation for small positive probabilities instead. Increase the displayed precision or use the more precise selected result.
When should I use a binomial distribution instead of Poisson?
Use a binomial distribution for a fixed number of independent trials with constant success probability. Poisson is commonly used to model event counts over a fixed interval at a constant average rate. Under suitable conditions, Poisson can approximate a binomial distribution.
Sources and methodology
NIST — Binomial Distribution defines the model and its probability functions. This table reuses our calculator’s log-gamma and log-sum-exp probability engine. Very small events can fall below IEEE-754 range; the display labels this limit explicitly.
Binomial Distribution Calculator handles ranges and complete calculations. Browse all tables or read our methodology.