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Probability Calculator

Find the union, intersection, and related probabilities of two independent events, solve from any two values, a series of repeats, or a normal-distribution area.

Union, intersection, and related probabilities of two independent events.

ExampleSample values — edit any field to see your result.

Probability that event A occurs. A probability between 0 and 1 (0 = impossible, 1 = certain).

Probability that event B occurs. A probability between 0 and 1 (0 = impossible, 1 = certain).

Results update as you type.

P(A ∪ B)

0.7

Estimated result

P(A ∪ B) — A or B or both
0.7
P(A ∩ B) — A and B both occur
0.2
P(A Δ B) — A or B but not both
0.5
P((A ∪ B)') — neither
0.3
P(A') — A does not occur
0.5
P(B') — B does not occur
0.6
A occurs but not B
0.3
B occurs but not A
0.2

All probabilities

EventProbability
P(A') — A does not occur0.5
P(B') — B does not occur0.6
P(A ∩ B) — A and B both occur0.2
P(A ∪ B) — A or B or both0.7
P(A Δ B) — A or B but not both0.5
P((A ∪ B)') — neither A nor B0.3
A occurs but not B0.3
B occurs but not A0.2

Calculation steps

QuantityFormulaResult
P(A')1 − P(A)0.5
P(B')1 − P(B)0.6
P(A ∩ B)P(A) × P(B)0.2
P(A ∪ B)P(A) + P(B) − P(A ∩ B)0.7
P(A Δ B)P(A) + P(B) − 2 P(A ∩ B)0.5
P((A ∪ B)')1 − P(A ∪ B)0.3
A but not BP(A) × (1 − P(B))0.3
B but not A(1 − P(A)) × P(B)0.2
P(A ∪ B)0.7

Find the union, intersection, exclusive or, and complements of two independent events. A solver fills in the rest from any two values. Separate tabs cover a series of independent repeats and the area under a normal curve.

Formula

Independent events multiply:

P(A ∩ B) = P(A) × P(B)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A Δ B) = P(A) + P(B) − 2 × P(A ∩ B)
P(A') = 1 − P(A)

A series of n independent trials of A has

P(all n succeed) = P(A)^n
P(none succeed) = (1 − P(A))^n
P(at least once) = 1 − (1 − P(A))^n

A normal random variable with mean μ and standard deviation σ has

z = (x − μ) / σ
P(Lb ≤ X ≤ Rb) = Φ(z_Rb) − Φ(z_Lb)

Type -inf or inf for an unbounded side.

The default two-event example is P(A) = 0.5, P(B) = 0.4. Then P(A ∩ B) = 0.2, P(A ∪ B) = 0.7, and P(A Δ B) = 0.5. The events are assumed independent — the calculator does not handle conditional probability P(A|B).

Examples

Two independent events

P(A) = 0.5 and P(B) = 0.4. P(A') = 0.5, P(B') = 0.6, P(A ∩ B) = 0.2, P(A ∪ B) = 0.7, P(A Δ B) = 0.5, neither = 0.3.

Any two values

Given P(A) = 0.5 and P(A ∩ B) = 0.4, independence gives P(B) = 0.4 / 0.5 = 0.8, then P(A ∪ B) = 0.9.

Series of repeats

P(A) = 0.6 five times and P(B) = 0.3 three times. P(A occurs all five times) = 0.6^5 = 0.07776. P(A occurs at least once) = 0.98976.

Normal area from -1 to 1

On the standard normal (μ = 0, σ = 1) the area between −1 and 1 is 0.68269. P(X ≤ −1) = P(X ≥ 1) = 0.15866. Type inf for +∞; the area from −∞ to 1 is 0.84134.

Frequently asked questions

How do I find the probability of A or B?
For independent events, P(A ∪ B) = P(A) + P(B) − P(A) × P(B). The default P(A) = 0.5 and P(B) = 0.4 gives 0.5 + 0.4 − 0.2 = 0.7.
What does independent mean?
Two events are independent when one does not change the chance of the other. Then P(A and B) is just the product P(A) × P(B). Drawing cards without replacement is not independent.
How does the two-event solver work?
Enter any two of P(A), P(B), the complements, the intersection, the union, the exclusive or, or neither. The calculator fills in the rest. Some pairs (intersection and union) have two (A, B) assignments.
What does the series tab calculate?
Event A is repeated n independent times and event B is repeated m times. "A occurs 5 times" is P(A)^5 (every trial succeeds). "A occurs at least once" is 1 − (1 − P(A))^n.
How is the normal-distribution area calculated?
Each bound is converted to a z-score (x − μ) / σ. The area between the bounds is Φ(z_right) − Φ(z_left), where Φ is the standard normal CDF. Type -inf or inf for an unbounded tail. The table lists μ ± zσ ranges for common confidence levels.

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