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Probability Calculator
Compute combined probabilities of two events, solve for unknown probabilities from any two inputs, evaluate repeated independent events, and find normal distribution probabilities.
Probability of Two Events
Enter P(A) and P(B) (values between 0 and 1) assuming A and B are independent.
Probability Solver for Two Events
Provide any 2 values below to calculate the rest. Does not assume independence.
Probability of a Series of Independent Events
Enter the probability of each event and the number of times it repeats.
| Probability | Repeat Times | |
|---|---|---|
| Event A | ||
| Event B |
Probability of a Normal Distribution
Compute the probability that a normal variable falls between two bounds, given mean and standard deviation.
Probability calculator guide
A probability calculator brings together the most common operations for reasoning about chance. It covers the probability of two events, solves for unknown probabilities when only a couple values are known, evaluates repeated independent trials, and estimates probabilities under the normal distribution.
The probability of two independent events assumes their outcomes do not influence each other. In that case P(A∩B) equals P(A) × P(B), and P(A∪B) simplifies to P(A) + P(B) − P(A)P(B). These formulas form the backbone of introductory probability courses and appear in puzzles, games, polling, and basic risk analysis.
The solver panel is more general because it does not assume independence. It accepts any two inputs from a standard set including P(A), P(B), P(A'), P(B'), P(A∩B), P(A∪B), P(AΔB), and P((A∪B)') and fills in the remaining values. It is useful when data is incomplete but some joint or complementary probabilities are known.
Repeated independent trials appear in reliability engineering, quality control, A/B tests, sports analytics, and many gameplay mechanics. Given a probability p and a repeat count n, the calculator reports the probability of all successes (pⁿ), at least one success (1 − (1 − p)ⁿ), and combined chances for two events.
Normal distribution probability is central to statistical inference and confidence interval reasoning. Given a mean, a standard deviation, and two bounds, the tool reports the probability that a random observation falls in the interval by computing standardized z-scores and the normal cumulative distribution function.
Use the steps panels to see each formula applied to your inputs. This is useful for homework, teaching, and spot-checking calculations.
How to use
- For two independent events, enter P(A) and P(B) and read the derived probabilities.
- For the solver, enter exactly two known probabilities and let the tool infer the rest.
- For a series of independent events, enter each probability and how many times it repeats.
- For the normal distribution, enter mean, standard deviation, and bounds to compute interval probability.
- Use the Steps panel to review the formulas applied.
Core formulas
Independent events: P(A∩B)=P(A)P(B); P(A∪B)=P(A)+P(B)−P(A)P(B); P(AΔB)=P(A)+P(B)−2P(A)P(B); P((A∪B)')=(1−P(A))(1−P(B)). General: P(A')=1−P(A); P(A∪B)=P(A)+P(B)−P(A∩B). Series of independent trials: P(all)=pⁿ; P(at least one)=1−(1−p)ⁿ. Normal: P(x₁<X<x₂)=Φ((x₂−μ)/σ)−Φ((x₁−μ)/σ).
Notes and limitations
- All probabilities must be between 0 and 1 inclusive.
- The independent section assumes A and B are independent; use the solver when that assumption does not hold.
- Normal distribution probabilities use a high-accuracy error function approximation.
- Solver expects exactly two inputs. Redundant or inconsistent inputs will raise a validation error.
FAQ
What does independent mean?
Two events are independent when the outcome of one has no effect on the probability of the other. Formally, P(A∩B) = P(A)P(B).
Why solve from only 2 inputs?
Because the 8 standard probabilities are linked by a small set of equations, any 2 generally determine the rest.
How accurate is the normal distribution?
It uses a rational approximation of the error function with double-precision accuracy sufficient for typical statistics work.
What does AΔB mean?
It is the symmetric difference: exactly one of A or B occurs, not both.