Binomial Distribution Calculator

Binomial Distribution Calculator – Exact Probabilities for Repeated Trials

The binomial distribution is one of the most widely used probability models in statistics. It describes the number of successes X in n independent trials, where each trial has only two outcomes (success or failure) and the probability of success p is the same each time.

The Binomial Distribution Calculator on MyTimeCalculator lets you explore this model numerically. You can compute exact probabilities for specific values of X, cumulative and tail probabilities for hypothesis tests, and a full table of P(X = k) for all possible outcomes.

How This Binomial Distribution Calculator Works

The calculator is organized into three modes that match common use cases:

• Single value probability: Compute P(X = k), P(X ≤ k), P(X ≥ k) and descriptive statistics for a specific k.
• Range & tail probability: Compute P(a ≤ X ≤ b), P(X ≤ k) or P(X ≥ k) along with complement probabilities.
• Binomial table: Generate a table of probabilities for k = 0, 1, 2, …, n with cumulative values.

All modes assume X follows a binomial distribution with parameters n (number of trials) and p (success probability): X ~ Binomial(n, p).

Mode 1: Single Value Probability (PMF, CDF & Tail)

In the single value tab you enter:

• The number of trials n
• The success probability p between 0 and 1
• A particular number of successes k

The calculator then returns:

• P(X = k), the probability of exactly k successes
• P(X ≤ k), the cumulative probability up to k
• P(X ≥ k), the upper tail probability
• Mean E[X] = np
• Variance Var(X) = np(1 − p)
• Standard deviation √(np(1 − p))

This is useful for homework problems, quality control scenarios and basic binomial hypothesis checks.

Mode 2: Range & Tail Probability

Many questions ask for the probability that X falls within an interval or in one tail of the distribution. In the range tab you choose a mode and specify one or two cut points:

• Between k1 and k2 (inclusive): Computes P(k1 ≤ X ≤ k2).
• At most k: Computes P(X ≤ k).
• At least k: Computes P(X ≥ k).

The calculator also reports the complement probability 1 − P(event), which is often useful for hypothesis tests and risk calculations.

Mode 3: Binomial Probability Table

The table tab generates a list of binomial probabilities for all possible values of X from 0 up to n. For each k it shows:

• P(X = k), the individual probability
• P(X ≤ k), the cumulative probability up to that point

The table is helpful when you want to see the whole shape of the distribution, check approximate symmetry, or look up multiple values without recalculating each one by hand. A maximum n is used to keep the table at a practical size.

Key Formulas for the Binomial Distribution

The binomial probability mass function and summary statistics can be written as:

• PMF: P(X = k) = C(n, k) p^k (1 − p)^{n − k} for k = 0, 1, …, n
• Mean: E[X] = np
• Variance: Var(X) = np(1 − p)
• Standard deviation: σ = √(np(1 − p))

Here C(n, k) = n! / (k!(n − k)!) is the binomial coefficient, sometimes read as “n choose k.” The calculator uses a numerically stable version of this coefficient to avoid overflow for moderate values of n.

When the Binomial Model is Appropriate

The binomial distribution is a good model under the following conditions:

• You have a fixed number of trials n.
• Each trial results in exactly two possible outcomes (often labelled success and failure).
• The probability of success p is the same on each trial.
• Trials are independent; the outcome of one does not affect the others.

Examples include the number of defective items in a batch, the number of heads in coin flips, the number of positive responses in a sample survey, or the number of successes in repeated experiments.

Tips for Using This Binomial Calculator

• Check that p is between 0 and 1 and that k lies between 0 and n.
• For very large n, probabilities can become extremely small; using more decimal places can reveal their magnitude.
• If you care mainly about whether X is unusually large or small, focus on tail probabilities P(X ≤ k) or P(X ≥ k).
• Use the table to see how probabilities are distributed across the possible values of X.

This calculator is intended for education and planning. It does not replace a full statistical analysis package, but it gives quick, exact numbers for many binomial problems you encounter in practice.