P(A)=10%, P(B)=50% — Probability

Full probability breakdown for two independent events. Use the calculator below to try your own values.

P(A and B) — Both Occur

Example

With P(A) = 10% and P(B) = 50% as independent events, there's a 5.0% chance both happen, a 55.0% chance at least one happens, and a 45.0% chance neither happens.

P(A or B) — At Least One

55%

P(A XOR B) — Exactly One

50%

P(Neither)

45%

P(A') / P(B')

90% / 50%

All possible outcomes

Both A and B: 5%
Only A: 5%
Only B: 45%
Neither: 45%

What Is Probability?

Probability measures how likely an event is to occur, expressed as a number from 0 (impossible) to 1 (certain) — or, equivalently, 0% to 100%. When two events are independent, the outcome of one has no effect on the odds of the other, which is the assumption this calculator makes about events A and B.

Every outcome for two independent events falls into exactly one of four categories: both A and B happen, only A happens, only B happens, or neither happens — and these four probabilities always add up to exactly 100%, which is what the pie chart above visualizes.

Key Probability Formulas

For independent events A and B: the complement is P(A') = 1 − P(A); the intersection (both occurring) is P(A ∩ B) = P(A) × P(B); the union (either or both) is P(A ∪ B) = P(A) + P(B) − P(A ∩ B); and the exclusive or (exactly one) is P(A ⊕ B) = P(A) + P(B) − 2 × P(A ∩ B).

Independent vs. Dependent Events

Two coin flips are independent — the first flip doesn't affect the second. Drawing two cards from a deck without replacement is dependent, since removing the first card changes the odds for the second. This calculator's multiplication shortcut for the intersection only holds for independent events; dependent events require conditional probability, P(A|B).

Mutually Exclusive Events

Mutually exclusive events can never both happen — P(A ∩ B) = 0 by definition — which is a different (and stronger) condition than independence. Rolling a 3 and rolling a 5 on the same single die roll are mutually exclusive; two separate coin flips landing heads are independent but not mutually exclusive.

Why the Four Outcomes Always Sum to 100%

"Both," "only A," "only B," and "neither" are mutually exclusive and collectively exhaustive — together they cover every possible combined outcome exactly once, so their probabilities must always total 1 (100%), regardless of the individual values of P(A) and P(B).

Example — Your Current Inputs

With P(A) = 10% and P(B) = 50% as independent events, there's a 5.0% chance both happen, a 55.0% chance at least one happens, and a 45.0% chance neither happens.

Additional Example — Two Coin Flips

With P(A) = 50% (first flip is heads) and P(B) = 50% (second flip is heads), there's a 25% chance both are heads, a 75% chance at least one is heads, a 50% chance exactly one is heads, and a 25% chance neither is heads.

About These Parameters

P(A) and P(B): The standalone probability of each event happening on its own, before considering the other event at all — for example, the odds a single product is defective, or the odds a single day sees rain.

Frequently Asked Questions

What if my events aren't independent?

This calculator's intersection formula (multiplying P(A) by P(B)) only applies to independent events. For dependent events, you'd need the conditional probability P(B|A) in place of P(B) in the intersection formula.

What's the difference between "or" and "XOR"?

P(A or B) — the union — includes the case where both happen. P(A XOR B) — exclusive or — only counts the case where exactly one happens, excluding the overlap.

Can P(A) and P(B) both be 100%?

Yes — if both events are certain, the intersection is also 100% and every other category (only A, only B, neither) is 0%.