The mechanics
When two events are independent—meaning one outcome doesn't influence the other—you can combine their individual probabilities using three core operations: AND (both occur), OR (at least one occurs), and NOT (an event doesn't happen). This calculator applies the mathematical rules for each.
The formula
P(A AND B) = P(A) × P(B) | P(A OR B) = P(A) + P(B) − P(A) × P(B) | P(NOT A) = 1 − P(A)
Worked example
Imagine you're checking weather forecasts. The probability of rain tomorrow is 0.4 (40%), and the probability of wind is 0.3 (30%). These are independent events.
Finding P(A AND B) — rain AND wind:
Multiply the two probabilities:
- P(rain AND wind) = 0.4 × 0.3 = 0.12
- As a percentage: 12%
There's a 12% chance both rain and wind occur together.
Finding P(A OR B) — rain OR wind (or both):
Use the formula: P(A) + P(B) − P(A) × P(B)
- P(rain OR wind) = 0.4 + 0.3 − (0.4 × 0.3)
- = 0.4 + 0.3 − 0.12
- = 0.58
- As a percentage: 58%
There's a 58% chance of at least one of these weather events. (We subtract the overlap—the 12%—to avoid counting it twice.)
Finding P(NOT A) — no rain:
Subtract from 1:
- P(NOT rain) = 1 − 0.4 = 0.6
- As a percentage: 60%
There's a 60% chance of no rain tomorrow.
Why the OR formula includes subtraction
When you add P(A) + P(B), you're counting the scenario where both happen twice—once in each probability. The term P(A) × P(B) represents that overlap, so you subtract it once to get the true probability of "at least one."
For example: if P(A) = 0.5 and P(B) = 0.5, adding them naively gives 1.0 (certainty), which is wrong. The correct answer is 0.5 + 0.5 − 0.25 = 0.75. There's a 75% chance at least one occurs, not 100%.
Common mistakes
Entering probabilities above 1 or below 0: Probabilities must always fall between 0 and 1 (or 0% and 100%). A value like 1.5 or −0.2 is impossible and will produce invalid results.
Confusing AND with OR: AND (both happen) produces smaller numbers; OR (at least one) produces larger ones. If you're unsure which you need, ask: "Do I want the chance of both, or the chance of at least one?"
Assuming events are independent when they're not: This calculator only works for truly independent events. If drawing a card changes the odds of the next draw, or if rain makes wind more likely, the formulas break down. You'd need conditional probability instead.
Forgetting to convert percentages: If you have probabilities as percentages (like 40%), convert them to decimals (0.4) before entering them into the calculator.