What Is Find Probability?
To find a probability you first choose the distribution that models your situation, then compute the area or sum that matches your question, such as P(X < x), P(X > x), P(X = x) or P(a ≤ X ≤ b).
Pick the distribution, enter its parameters and the value x, and the calculator returns the probability and its complement, the mean and standard deviation, and a chart with the region highlighted. For discrete distributions it distinguishes < from ≤; for continuous ones it explains that a single value has probability zero.
The article below includes a table to help you choose the right distribution, the key formulas, worked examples and common questions.
How to Use the Find Probability Calculator
- Pick the distribution and enter its parameters.
- Choose the type of probability.
- Enter x (and x₂ for a range).
- Read the probability and its complement and check the chart.
Find Probability Formula
Range probability
Upper tail
Standardising (normal)
Find Probability Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| X | The random variable | |
| x | The value in your question | |
| F(x) | Cumulative probability P(X ≤ x) | |
| Parameters | μ, σ, n, p, λ, a, b or df | depend on the distribution |
Find Probability Worked Example
With the example values loaded in the calculator above, the steps are:
- Model: Normal(100, 15²).
- P(X ≤ 115) = 0.84134475.
- For a continuous distribution P(X < x) = P(X ≤ x), because single points have zero probability.
Result: P(X ≤ 115) = 0.84134475.
How to Read a Probability Result
Choose a distribution, enter its parameters, and pick the kind of probability: at most, at least, exactly, or between two values. The result is a number from 0 to 1, with its complement shown next to it.
The complement 1 − P answers the opposite question. If P(X ≤ 115) is the share at or below 115, then the complement is the share above 115. The calculator also gives the mean and standard deviation of the chosen distribution so you can judge how far your value sits from the centre.
Worked example
For scores with mean 100 and SD 15, P(X ≤ 115) = Φ((115 − 100) ÷ 15) = Φ(1) = 0.8413. For a binomial with n = 20 and p = 0.3, P(X ≤ 6) = 0.6080.
Where Find Probability Is Used
Normal probabilities
Find the chance that a normally distributed value falls below, above or between limits.
Counts
Use the binomial or Poisson for counts of successes or events.
Waiting times
Use the uniform or exponential for times.
Small samples
Use the t distribution when the standard deviation is estimated.
Common Find Probability Mistakes to Avoid
- Match the distribution to the situation. Counts of successes with fixed n suit the binomial, and events per interval suit the Poisson.
- For discrete distributions, P(X ≤ k) and P(X < k) differ. Check the wording.
- "At least" means ≥ and "more than" means >.
- Check that parameters make sense, such as p between 0 and 1.
Find Probability Terms and Definitions
- CDF
- The cumulative distribution function F(x) = P(X ≤ x).
- PMF
- The probability mass function for a discrete variable.
Find Probability FAQs
How do I find a probability?
Pick the distribution, enter its parameters, and choose whether you want a probability below, above, between or exactly at a value. The calculator returns it with its complement.
How do I find the probability between two values?
Subtract the cumulative probability at the lower value from the one at the upper value: P(a ≤ X ≤ b) = F(b) − F(a) for a continuous variable.
What is the probability of exactly one value?
For a continuous distribution it is 0. For a discrete one, such as the binomial or Poisson, it is the probability mass at that value.
How do I find "at least" probability?
Compute 1 minus the probability of fewer than that value. For a discrete variable, P(X ≥ k) = 1 − P(X ≤ k − 1).