Binomial Probability Calculator (With Step-by-Step Solutions)
Calculate single event, combined, conditional and complementary probabilities instantly. Pro Mode adds binomial distribution, normal distribution, step-by-step working and history.
Probability of getting exactly k successes in n independent trials, each with success probability p.
Calculates probability that a normally distributed variable X falls in a given range using the Z-score method.
Calculates cumulative binomial probabilities — at least k successes, at most k, or exactly k in n trials.
Calculate combinations C(n,r) and permutations P(n,r) — the number of ways to choose or arrange items.
Free Binomial Probability Calculator — Single Event to Advanced Distributions
Probability is the mathematical language of uncertainty — a precise framework for quantifying how likely an outcome is before it occurs. This browser-based binomial probability calculator handles the full spectrum of probability problems: from the simplest single-event questions ("what is the probability of rolling a 6?") through to multi-step discrete distributions, conditional reasoning and continuous normal distribution queries. Every calculation is performed entirely in your browser with no data sent to any server, no account required and no usage limits.
Students, researchers and professionals encounter probability in examinations, data analysis, quality control, medical research and financial modelling. Whether you need a quick complement check or a comprehensive binomial probability distribution calculator for discrete parameter problems, all modes are available in a single tool — toggle between Simple and Pro for the level of detail you need.
Formula Reference — The Equations Behind Every Calculation
Note on formula display: All equations above are rendered as native HTML text, not images or canvas elements, so search engines can read and index every formula directly. This is intentional — mathematical content written in text carries full keyword weight and is accessible to screen readers.
Binomial Probability Calculator with Steps — Three Worked Examples
The fastest way to understand a probability tool is to see it solve a real problem. The three examples below mirror the sample calculations built directly into the tool. Switch to Pro mode and click any sample button to load these values automatically, then see the full working panel underneath the result. This is the same structured approach expected in A-level Statistics, university probability modules and professional certification exams.
Each of these worked solutions is reproduced step-by-step inside the tool's Pro mode working panel. The panel shows the formula, the substitution of actual values, each arithmetic stage and the interpreted result — structured exactly as university and A-level mark schemes require written working to be presented.
The Math Behind Our Binomial Probability Formula Calculator
The binomial distribution models situations where you repeat the same experiment n times, each trial is independent of the others, and each trial can only produce one of two outcomes — success or failure — with a fixed probability p of success. It is one of the most widely applied distributions in statistics: quality testing (how many defective units in a batch?), medical research (how many patients respond to treatment?), and even sports analytics (how many goals in a season?).
The core binomial probability formula is P(X=k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ. It has three components working together. The combination term C(n,k) counts the number of distinct arrangements of k successes within n trials — this is why the formula is sometimes written with the binomial coefficient symbol. The term pᵏ is the probability that all k successes occur. The term (1−p)ⁿ⁻ᵏ is the probability that all remaining trials result in failure. Multiplied together with C(n,k) to account for every possible ordering, they give the total probability of exactly k successes.
Cumulative Binomial Probability Calculator — At Least and At Most
Many real-world questions are not about exact counts but thresholds: "what is the probability of getting at least 2 sixes in 5 rolls?" or "what is the chance of at most 1 failure in 8 trials?" These require summing across multiple terms of the binomial distribution — this is what a cumulative binomial probability calculator does.
The cumulative binomial probability calculator mode on this tool handles all three cases. For P(X ≥ k), it sums the binomial PMF from i = k to n. For P(X ≤ k), it sums from i = 0 to k. Both directions are computed in real time from the same n, p and k inputs — just select the type from the dropdown. The step-by-step panel describes which terms are being summed and why, which is particularly useful when revising for statistics exams where method marks depend on correct identification of the summation range.
| Calculator Mode | Formula | Example Question | When to Use |
|---|---|---|---|
| Single Event | P(A) = f ÷ n | Probability of drawing an Ace from a 52-card deck | One trial, classical probability |
| Complement | P(A') = 1 − P(A) | Probability it does NOT rain given P(rain) = 0.3 | Finding the "opposite" probability |
| Conditional | P(A|B) = P(A∩B) ÷ P(B) | P(Pass | Studied) given joint and marginal probabilities | Dependent events, Bayesian reasoning |
| Multiple Events | P(A∪B) = P(A)+P(B)−P(A∩B) | Probability of rolling even OR greater than 4 on a die | OR / AND questions with two events |
| Binomial (exact) | C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ | Exactly 3 heads in 5 fair coin flips | Fixed trials, two outcomes, exact count |
| Cumulative Binomial | Σ P(X=i) for range | At least 2 sixes in 5 dice rolls | Threshold questions: at least / at most |
| Normal Distribution | Φ((x−μ)÷σ) | P(IQ < 115) with μ=100, σ=15 | Continuous data, bell-curve questions |
| Combinations / Perms | C(n,r) and P(n,r) | Ways to choose 6 numbers from 49 (lottery) | Counting arrangements and selections |
This tool functions as a probability calculator with steps across all eight modes — not just a numeric answer but a structured working panel showing the formula, substitution and each arithmetic stage. In Pro mode, every calculation produces step-by-step output that matches the written working format expected in GCSE, A-level and undergraduate statistics assessments.
Frequently Asked Questions
Common questions about binomial probability, distributions and calculator use
A binomial probability calculator computes P(X=k) — the probability of getting exactly k successes in n independent trials where each trial has a fixed success probability p. It applies the formula P(X=k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ and also returns the mean (μ = np), variance (σ² = np(1−p)) and standard deviation of the distribution. This tool covers binomial alongside seven other probability modes in one place.
Binomial probability gives the chance of exactly k successes. The cumulative binomial probability calculator mode sums across a range: P(X ≤ k) adds terms from 0 to k, while P(X ≥ k) adds from k to n. Use the exact mode for "precisely k" questions and the cumulative mode for "at least" or "at most" threshold questions — all three options are available from the dropdown in the At Least / At Most tab.
Switch to Pro mode and open the Binomial tab. Enter n (total trials), k (successes wanted) and p (probability per trial), then click Calculate. A numbered step-by-step panel appears below the result showing: identifying values, the formula, computing C(n,k), substituting all values and the final answer. This binomial probability calculator with steps output mirrors the written working required in exam mark schemes.
Conditional probability P(A|B) is the probability of A occurring given that B has already happened. The formula is P(A|B) = P(A∩B) ÷ P(B). You supply the joint probability P(A∩B) and the condition probability P(B) — P(B) must be greater than zero. The calculator returns P(A|B) with a bar chart, percentage and full step-by-step working panel in Pro mode.
Enter the mean μ, standard deviation σ and your value x. The calculator computes Z = (x − μ) ÷ σ to standardise the variable, then applies the cumulative normal distribution Φ(Z). You can find P(X < x), P(X > x), or P(x₁ < X < x₂) for between-range queries. The tool uses the Abramowitz and Stegun polynomial approximation, accurate to ±1.5×10⁻⁷, equivalent to a 7-decimal-place Z-table.
The binomial probability distribution calculator is used wherever you have a fixed number of independent trials with a constant success probability: coin flips, dice rolls, pass/fail quality testing, medical treatment response rates, survey responses and sports statistics. The distribution returns both individual probabilities P(X=k) and summary statistics — mean, variance and standard deviation — giving a complete picture of the expected behaviour across all n trials.
Combinations C(n,r) count selections where order does not matter: C(n,r) = n! ÷ (r! × (n−r)!). Permutations P(n,r) count arrangements where order matters: P(n,r) = n! ÷ (n−r)!. The relationship is P(n,r) = C(n,r) × r! — multiplying combinations by the number of ways r items can be ordered. The tool calculates both simultaneously and shows the step-by-step derivation for each.
Yes — Pro mode unlocks full step-by-step working on all eight calculation types. Each panel shows the formula, value substitution and every arithmetic stage in numbered steps, mirroring the written working format expected in GCSE, A-level and undergraduate probability exams. The sample buttons load real worked examples (rolling a die, coin flips, IQ distribution, lottery combinations) so you can immediately see how each type of problem is approached and solved.