Probability Calculator: And, Or, Given & At Least Once

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Formula at a glance

Probability Calculator: Addition rule — P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Example: Independent events 50% and 40% → P(A or B) — union: 0.7 (70%) — How the Probability Calculator works: Addition rule — P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

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📘 About the Probability Calculator

Probability measures how likely an event is, from 0 (impossible) to 1 (certain). With two events you often need the chance that both happen (intersection), that at least one happens (union), that one happens given the other, or that neither happens.

Enter P(A) and P(B) and say whether the events are independent, mutually exclusive or have a known overlap. The calculator applies the addition, multiplication and conditional probability rules and shows every combination, plus the chance of A happening at least once in repeated tries.

🧮 Formula

Addition rule

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Independent

P(A ∩ B) = P(A)·P(B)

Conditional

P(A | B) = P(A ∩ B) / P(B)

At least once in n tries

1 − (1 − p)ⁿ

✏️ Worked examples

Independent events 50% and 40%

You enter
P(A)	0.5
P(B)	0.4
A and B are	Independent

You get
P(A or B) — union	0.7 (70%)
P(A and B) — intersection	0.2 (20%)
P(not A) · P(not B)	0.5 · 0.6
P(neither A nor B)	0.3 (30%)
P(exactly one of A, B)	0.5 (50%)
P(A \| B) — A given B	0.5 (50%)
P(B \| A) — B given A	0.4 (40%)
Independent?	Yes — P(A and B) = P(A)·P(B)

0.7 (70%).

Rolling a six at least once in 4 throws

You enter
P(A)	1/6
P(B)	0.5
A and B are	Independent
Repeat A this many times (optional)	4

You get
P(A or B) — union	0.583333 (58.3333%)
P(A and B) — intersection	0.0833333 (8.33333%)
P(not A) · P(not B)	0.833333 · 0.5
P(neither A nor B)	0.416667 (41.6667%)
P(exactly one of A, B)	0.5 (50%)
P(A \| B) — A given B	0.166667 (16.6667%)
P(B \| A) — B given A	0.5 (50%)
Independent?	Yes — P(A and B) = P(A)·P(B)
A at least once in 4 independent tries	0.517747 (51.7747%)

0.517747 (51.7747%).

🧭 How to use this calculator

What you enter

P(A)0.5, 50% or 1/2
P(B)
A and B are
P(A and B) (if known)
Repeat A this many times (optional)probability A happens at least once

What you get

P(A or B) — union
P(A and B) — intersection
P(not A) · P(not B)
P(neither A nor B)
P(exactly one of A, B)
P(A | B) — A given B
P(B | A) — B given A
Independent?

❓ Frequently asked questions

What is the difference between independent and mutually exclusive?

Independent events do not affect each other (two coin tosses). Mutually exclusive events cannot happen together (rolling a 2 and a 5 on one die).

How do I find the probability of A or B?

Add their probabilities and subtract the overlap: P(A) + P(B) − P(A and B).

What is the chance of at least one six in four rolls?

1 − (5/6)⁴ ≈ 51.8%.

🎲 Probability Calculator (Two Events: And, Or, Given)