Probability Calculator
Work with probabilities four ways — combine two independent events, solve for unknowns from any two knowns, chain a series of independent events, or find the probability a normal-distribution value lands in a range, with a shaded bell curve.
1. Probability of Two Events
Assumes A and B are independent, so P(A ∩ B) = P(A) · P(B). For dependent events, use the solver below and enter P(A ∩ B) directly.
2. Probability Solver for Two Events
Fill in any two known values and the solver fills in the rest using the complement, union, and intersection identities.
3. Probability of a Series of Independent Events
Enter one row per event with its probability (0–1) and how many times it repeats. The combined probability is the product of each pᵢ raised to its repeat count.
4. Probability of a Normal Distribution
Enter the mean μ, standard deviation σ, and the range [left, right]. Use -inf or inf for an unbounded side. The tool computes P(left ≤ X ≤ right) exactly and shows the corresponding Z-scores — no manual Z-table needed.
What is probability?
Probability measures how likely an event is, on a scale from 0 ("this will not happen") to 1 ("this is certain"). A fair coin landing heads has probability 0.5. Probabilities can also be written as percentages (0.5 = 50%) or as fractions (1/2).
Probability rules explained, rule by rule
Each card below covers one core probability rule — the complement, intersection, union, XOR and normal-distribution area — with a plain definition, its formula, a diagram, and a worked example.
Complement — probability of NOT
The complement of A is 'A does not happen'. Its probability is one minus P(A). This one-line rule is the fastest way to compute 'at least one' problems: P(at least one) = 1 − P(none).
Intersection — both A and B (independent)
When two events are independent (knowing one doesn't change the other), the probability that both happen is the product of the two probabilities. Independence is a stronger condition than most people assume — check before you multiply.
Mutually exclusive events
Mutually exclusive means A and B cannot both happen on the same trial — for example, rolling a 2 or a 5 on a single die. The intersection is zero, and P(A or B) simplifies to P(A) + P(B). Mutually exclusive is not the same as independent.
Union — A or B (or both)
The union counts the ways A happens, B happens, or both. Inclusion–exclusion subtracts the overlap so it's not double-counted. When the events are mutually exclusive the overlap is zero and the formula collapses to P(A) + P(B).
Exclusive OR — exactly one of A or B
XOR is 'A or B, but not both'. You take the union and subtract twice the intersection, because the both-happen case is counted once in each of P(A) and P(B) and must be removed entirely.
Normal distribution — area under the bell curve
For a normally-distributed variable, the probability of landing in a range [L, R] is the shaded area under the bell curve between those bounds. Convert L and R into Z-scores and subtract the two CDF values.
Common mistakes
- Confusing independent with mutually exclusive. Independent events usually have overlap (P(A ∩ B) > 0). Mutually exclusive events cannot overlap (P(A ∩ B) = 0).
- Assuming independence when events are actually linked. Drawing a card without replacement is not independent.
- Forgetting to subtract the intersection in the union formula. P(A ∪ B) = P(A) + P(B) only if A and B cannot both happen.
- Adding probabilities greater than 1. If your total exceeds 1, you have almost certainly double-counted an overlap.
- Reading a Z-table for the wrong tail. This calculator sidesteps the ambiguity by always computing P(L ≤ X ≤ R) directly.
Features of this calculator
- Four tools on one page: independent-events combiner, unknown-value solver, series-of-events chain, and normal-distribution probability with a shaded bell curve.
- Solver derives every missing value from as few as two knowns using the complement, union, and intersection identities — flags contradictions instead of silently rounding.
- Normal-distribution tool accepts
-infandinfbounds and computes exact probabilities via the erf function. - Venn diagram gallery and bell curve rendered with semantic design tokens — follow your light/dark theme automatically.
- Independence assumption stated explicitly on the two-event tool.
Frequently asked questions
+What's the difference between independent and mutually exclusive events?
Independent events don't influence each other — P(A ∩ B) = P(A)·P(B). Mutually exclusive events cannot both happen — P(A ∩ B) = 0.
+How do I calculate the probability of an event NOT happening?
Subtract from 1. P(A') = 1 − P(A). If P(rain) = 0.3, then P(no rain) = 0.7.
+Do I need to enter probabilities as decimals?
Yes — use values between 0 and 1 (e.g. 0.25, not 25%).
+How does the solver decide which values to derive?
It repeatedly applies the complement and inclusion–exclusion identities until nothing new can be derived. Contradictions are reported rather than silently rounded.
+Can the normal-distribution tool handle one-sided ranges?
Yes. Use -inf for the left bound to get P(X ≤ R), or inf for the right bound to get P(X ≥ L).