How to find probability
P(Event) = Favorable Outcomes ÷ Total Outcomes Example: drawing an ace from a standard 52-card deck P(Ace) = 4 ÷ 52 = 1⁄13 = 7.69% "Favorable" just means the outcomes you're asking about, not that they're good news — the probability of rolling a 1 on a die is a "favorable outcome" in this sense even if you were hoping for a 6.
Combining two events
P(A or B) = P(A) + P(B) − P(A and B) The subtraction matters. Simply adding P(A) and P(B) double-counts the outcomes where both happen at once — P(A and B) gets subtracted back out exactly once to correct for that. How you find P(A and B) depends on the relationship between the two events.
Independent events
Two events are independent when one happening tells you nothing about whether the other does — separate dice rolls, separate coin flips, unrelated events.
P(A) = 1⁄6, P(B) = 1⁄6 P(A and B) = P(A) × P(B) = 1⁄6 × 1⁄6 = 1⁄36 ≈ 2.78% P(A or B) = 1⁄6 + 1⁄6 − 1⁄36 = 11⁄36 ≈ 30.56% Mutually exclusive events
Two events are mutually exclusive when they can't both happen — a single card drawn can't be both a King and a Queen.
P(King) = 4⁄52, P(Queen) = 4⁄52 P(King and Queen) = 0 (impossible for one card) P(King or Queen) = 4⁄52 + 4⁄52 − 0 = 8⁄52 ≈ 15.38% Dependent events
When events influence each other, P(A and B) has to be known or measured directly — it isn't derivable from P(A) and P(B) alone.
P(plays a sport) = 40%, P(plays an instrument) = 30% P(plays both) = 15% (measured directly, not calculated) P(plays a sport or an instrument) = 40% + 30% − 15% = 55% Why P(A and B) can't be just any number
Given P(A) and P(B), P(A and B) is boxed in on both sides. It can never exceed the smaller of the two individual probabilities — the overlap can't be bigger than either circle in a Venn diagram. And if P(A) and P(B) are both large, they're forced to overlap by at least a certain amount, or P(A or B) would exceed 100%, which is impossible.
Minimum possible overlap: P(A) + P(B) − 100% = 70% + 50% − 100% = 20% These two events cannot overlap by less than 20%, no matter what they are. This calculator checks both bounds on the Custom option in the Two Events tab, so an impossible combination gets caught with a clear explanation rather than quietly producing a result that doesn't add up.
Frequently asked questions
How do you calculate probability?
Divide the number of favorable outcomes by the total number of possible outcomes. Rolling a 3 on a six-sided die has 1 favorable outcome out of 6 total, so the probability is 1⁄6, about 16.67%.
What's the difference between independent and mutually exclusive events?
Independent events don't affect each other's probability at all — both can still happen together, like two separate coin flips both landing heads. Mutually exclusive events can never happen together — a single die roll can't show both a 2 and a 5. They're opposite ideas, not variations of the same thing.
How do you find the probability of A or B?
Add P(A) and P(B), then subtract P(A and B) to remove the double-counted overlap. For independent events, P(A and B) is P(A) × P(B); for mutually exclusive events, it's 0; otherwise it needs to be known directly.
What is the complement of an event?
The complement is "the event doesn't happen." It always equals 100% minus the event's own probability, since together they cover every possibility — if there's a 30% chance of rain, there's a 70% chance it doesn't rain.