How this calculator works
This calculator covers three common probability questions: the probability of a single event from a count of favorable and total outcomes, the combined probability of two independent events happening together ("and") or either one happening ("or"), and the complement of an event, the probability it does not happen.
Every probability here is a value between 0 (impossible) and 1 (certain), shown as a percentage. The two-event mode assumes independence, meaning the outcome of one event has no effect on the other, which is true for separate dice rolls or coin flips but not for events like drawing cards from a deck without replacement.
The formulas
Single event: P = favorable outcomes ÷ total outcomesTwo independent events, both occur: P(A and B) = P(A) × P(B)Two independent events, at least one occurs: P(A or B) = P(A) + P(B) − P(A) × P(B)Complement: P(not A) = 1 − P(A)The P(A or B) formula subtracts P(A and B) to avoid double-counting the outcomes where both events happen, which would otherwise be added in twice.
Worked example: coin flip and die roll together
- Event A: flipping heads on a fair coin, P(A) = 0.5.
- Event B: rolling a 6 on a fair 6-sided die, P(B) = 1/6 ≈ 0.1667.
- Both happen (independent): P(A and B) = 0.5 × 0.1667 ≈ 0.0833, about 8.33%.
- At least one happens: P(A or B) = 0.5 + 0.1667 − 0.0833 = 0.5834, about 58.34%.
Frequently asked questions
What does "independent events" mean?
Two events are independent when the outcome of one has no effect on the probability of the other. Separate coin flips and dice rolls are independent; drawing two cards from a deck without putting the first one back is not, since removing a card changes the odds for the second draw.
How is probability different from odds?
Probability is expressed as a fraction or percentage of all outcomes, like 1/6 or about 16.67%. Odds compare favorable to unfavorable outcomes directly, often phrased as "1 in 6", meaning 1 favorable outcome for every 6 total attempts.
Why subtract P(A and B) when calculating P(A or B)?
Without subtracting, the outcomes where both A and B happen would be counted twice — once inside P(A) and again inside P(B). Subtracting P(A and B) corrects for that overlap.
What is the complement of an event used for?
The complement is often easier to calculate than the event itself. For example, "at least one of 3 coin flips is heads" is easier to solve as 1 minus "all 3 flips are tails".
Can probability be negative or greater than 1?
No. A valid probability is always between 0 (the event never happens) and 1 (the event always happens), inclusive. This calculator rejects inputs outside that range.