A clear answer before the details
Probability is a number from 0 to 1, or 0% to 100%, describing how likely an event is. The correct formula depends on the experiment and assumptions.
Choose the probability model that fits
A binomial probability calculator finds the chance of exactly k successes in n independent trials when each trial has the same success probability p. The formula is P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ. Use a conditional probability calculator for P(A|B), a dice probability calculator for target sums, or a normal distribution calculator for a continuous value described by a mean and standard deviation.
Interpret probability as a model, not a promise
Probability is between 0 and 1, or 0% and 100%. A result such as 0.0439453125 means the model assigns about a 4.3945% chance to that event under the stated assumptions. Check that k is not greater than n, every probability is valid, and the conditioning event is not zero. In real applications, independence, sample size, and the chosen distribution determine whether the answer is a useful approximation.
Formula guide
The key equations are separated below. Start with the calculator mode, then choose the shape or model that matches your problem.
Binomial probability
Use equal-probability independent trials.
Conditional probability
The condition P(B) must be greater than zero.
Dice probability
n dice with s sides produce sⁿ equally likely outcomes.
Normal distribution
Standardise x before reading a normal probability.
Normal cumulative probability
Φ(z) is the area to the left of z under the standard normal curve.
Worked example
For n = 10, k = 2, and p = 0.5, P(X = 2) = C(10,2)(0.5)²(0.5)⁸ = 0.0439453125.
Formulas for common use cases
Use the matching equation for the problem you are solving, then enter its variables in the calculator above.
Dice target sums
The calculator counts each equally likely dice outcome.
At least one success
The complement is often easier to calculate.
Expected value
Multiply each outcome by its probability and add.
How to use this calculator
- Choose binomial, conditional, dice, or normal mode.
- Enter values within their valid ranges.
- Review the decimal, percentage, and formula steps.
- Use the result as a mathematical model, not a prediction of a specific outcome.
Where it helps
These models support classroom probability, quality-control sampling, dice and game mathematics, and introductory statistics. The tool explains the assumptions so you can tell when a binomial model is appropriate or when a normal distribution is only an approximation.
Common mistakes
Probabilities must stay between 0 and 1. In binomial mode, k cannot exceed n. Standard deviation must be positive, and conditional probability requires a non-zero conditioning event. The calculator does not make gambling, investment, or real-world outcome predictions.