Probability Distribution Calculator
Explore normal, binomial, Poisson, uniform, exponential, geometric, hypergeometric, t, chi-square, and F distributions in one student-friendly calculator. Use it as a PDF, CDF, PMF, left-tail, right-tail, between-values, and exact-probability calculator for common statistics distributions. Calculate probabilities, view shaded graphs, compare distributions, and learn what each result means step by step.
Background
Probability distributions describe how likely different outcomes are. Some distributions are continuous, like the normal distribution, while others are discrete, like the binomial or Poisson distribution. This calculator helps students connect formulas, graphs, probability notation, and real-world interpretation.
How to use this calculator
- Choose the distribution that matches your problem.
- Enter the required parameters, such as mean and standard deviation for a normal distribution.
- Choose the probability type, such as P(X ≤ x) or P(a ≤ X ≤ b).
- Click Calculate to see the probability, graph, formula, table, and interpretation.
- Use the quick picks to explore common student examples instantly.
How this calculator works
- For continuous distributions, it calculates area under the curve.
- For discrete distributions, it calculates sums of probability mass values.
- It shades the graph to show the exact probability region.
- It reports key statistics such as mean, variance, and standard deviation.
- It explains the result in plain language so students understand what the probability means.
Formula & Equations Used
Normal distribution: f(x) = 1/(σ√(2π)) · e^(-(x - μ)²/(2σ²))
Binomial distribution: P(X = k) = C(n,k)p^k(1-p)^(n-k)
Poisson distribution: P(X = k) = e^-λ λ^k / k!
Uniform distribution: f(x) = 1/(b - a)
Exponential distribution: f(x) = λe^(-λx)
Geometric distribution: P(X = k) = (1-p)^(k-1)p
Hypergeometric distribution: P(X = k) = [C(K,k)C(N-K,n-k)] / C(N,n)
Example Problem & Step-by-Step Solution
Example 1 — Normal distribution
- A test has mean μ = 75 and standard deviation σ = 10.
- Find the probability that a student scores at most 85.
- Choose Normal distribution.
- Enter mean = 75 and standard deviation = 10.
- Choose P(X ≤ x) and enter x = 85.
- The calculator converts the value to a z-score and finds the cumulative probability.
Example 2 — Binomial distribution
- A quiz has 10 true/false questions.
- A student guesses randomly, so p = 0.5.
- Find the probability of getting exactly 7 correct.
- Choose Binomial distribution.
- Enter n = 10, p = 0.5, and x = 7.
- Choose exact probability.
Example 3 — Poisson distribution
- A help desk receives an average of 4 calls per hour.
- Find the probability of receiving 6 or fewer calls in an hour.
- Choose Poisson distribution.
- Enter λ = 4 and x = 6.
- Choose P(X ≤ x).
Frequently Asked Questions
Q: Which distribution should I choose?
Use normal for bell-shaped continuous data, binomial for a fixed number of success/failure trials, Poisson for event counts over time or space, and uniform when all values in an interval are equally likely.
Q: What is the difference between discrete and continuous distributions?
Discrete distributions count separate outcomes, such as 0, 1, 2, or 3 successes. Continuous distributions describe measurements across an interval, such as height, time, or score.
Q: Can this calculator show shaded probability regions?
Yes. The graph shades the region matching the selected probability type.
Q: Is this useful for statistics homework?
Yes. It is designed to help students calculate probabilities, understand distribution shapes, and explain answers clearly.
Q: Should Pearson also create separate calculators for each distribution?
Yes. This flagship calculator is ideal as a hub, while separate distribution calculators are better for focused SEO and exact student search intent.
All calculators