Understanding the four results
The table below shows each output for the example P(A) = 50%, P(B) = 30%, assuming independence.
| Outcome | Formula | Example value |
|---|---|---|
| Both A and B | P(A) x P(B) | 0.5 x 0.3 = 15% |
| At least one (A or B) | P(A) + P(B) - P(A)P(B) | 0.5 + 0.3 - 0.15 = 65% |
| A does not occur | 1 - P(A) | 1 - 0.5 = 50% |
| Neither occurs | (1 - P(A)) x (1 - P(B)) | 0.5 x 0.7 = 35% |
- All results assume A and B are independent. If the events influence each other (dependent events), these formulas do not apply and conditional probabilities are required.
- 'At least one occurs' and 'neither occurs' are complementary: their probabilities always sum to 100%.
- Probabilities describe long-run frequencies over many repetitions; a 15% chance does not mean the event occurs in exactly 15 of the next 100 trials.
What is the probability of combined events?
A probability is a number between 0 and 1 (or 0% and 100%) expressing how likely an event is: 0 means impossible, 1 means certain. When two events are considered together, the standard rules of probability determine how their individual probabilities combine into the probability of joint outcomes such as 'both occur' or 'at least one occurs'.
This calculator assumes the two events are independent: knowing that one occurred tells you nothing about whether the other occurs. Coin flips, separate dice rolls, and unrelated random draws are classic examples of independence. Under independence, the probability that both events occur is simply the product of their individual probabilities: P(A and B) = P(A) x P(B).
The independence assumption is essential and must be checked against the real situation, not assumed by default. Drawing two cards from the same deck without replacement, or two weather events on the same day, are dependent — the first outcome changes the probabilities for the second — and the product rule then understates or overstates the true joint probability. For dependent events, conditional probabilities are required: P(A and B) = P(A) x P(B given A).
How to use this probability calculator
- Enter the probability of event A as a percentage between 0 and 100, for example 50 for a fair coin landing heads.
- Enter the probability of event B as a percentage, for example 30.
- Confirm that the two events are genuinely independent — one occurring must not change the chance of the other. The formulas used here are only valid under that assumption.
- Read the four combined probabilities: both occur, at least one occurs, A does not occur, and neither occurs.
Probability rules for independent events
Multiplication rule (both occur): for independent events, P(A and B) = P(A) x P(B). Complement rule: P(not A) = 1 - P(A). Neither occurs: P(neither) = (1 - P(A)) x (1 - P(B)), which applies the multiplication rule to the two complements. At least one occurs (inclusion-exclusion): P(A or B) = P(A) + P(B) - P(A and B).
Worked example with P(A) = 50% and P(B) = 30%, written as 0.5 and 0.3. Both: 0.5 x 0.3 = 0.15 = 15%. At least one: 0.5 + 0.3 - 0.15 = 0.65 = 65%. Not A: 1 - 0.5 = 0.5 = 50%. Neither: 0.5 x 0.7 = 0.35 = 35%. As a consistency check, 'at least one' and 'neither' are complements: 65% + 35% = 100%.
The subtraction of P(A and B) in the 'at least one' formula prevents double counting: outcomes where both events occur would otherwise be counted once in P(A) and again in P(B).
Common mistakes
- Adding probabilities for 'A or B' without subtracting the overlap: P(A) + P(B) alone double-counts outcomes where both occur, and can even exceed 100%.
- Multiplying probabilities of dependent events as if they were independent — drawing cards without replacement is the classic counterexample.
- The gambler's fallacy: treating independent trials as if past outcomes change future probabilities; a fair coin that landed heads five times still has a 50% chance of heads on the next flip.
- Confusing 'both occur' with 'at least one occurs' — for P(A) = 50% and P(B) = 30% these are 15% and 65%, very different quantities.
Perguntas frequentes
What does it mean for two events to be independent?
Two events are independent when the occurrence of one does not change the probability of the other — formally, P(A and B) = P(A) x P(B). Separate coin flips and unrelated dice rolls are independent; drawing two cards from the same deck without replacement is not, because the first draw changes the composition of the deck.
How do I calculate the probability that both events happen?
For independent events, multiply the individual probabilities: P(A and B) = P(A) x P(B). For example, with P(A) = 50% and P(B) = 30%, the probability that both occur is 0.5 x 0.3 = 0.15, or 15%. This product rule is only valid when the events are independent.
How do I calculate the probability that at least one event happens?
Use the inclusion-exclusion rule: P(A or B) = P(A) + P(B) - P(A and B). With independent events at 50% and 30%, that is 0.5 + 0.3 - 0.15 = 0.65, or 65%. Equivalently, compute 1 minus the probability that neither occurs: 1 - (0.5 x 0.7) = 0.65.
Why can't I just add the two probabilities for 'A or B'?
Because outcomes where both events occur would be counted twice — once in P(A) and once in P(B). Adding 50% and 30% gives 80%, but the correct answer for independent events is 65% once the 15% overlap is subtracted. Simple addition is valid only for mutually exclusive events, which by definition cannot both occur.
What if my events are not independent?
Then these formulas do not apply. For dependent events, the joint probability uses a conditional probability: P(A and B) = P(A) x P(B given A), where P(B given A) is the probability of B once A is known to have occurred. For example, the chance of drawing two aces from a standard deck without replacement is (4/52) x (3/51), not (4/52) x (4/52).
Referências
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods (probability foundations). nist.gov.
- Ross SM. A First Course in Probability. Pearson (independence, inclusion-exclusion and conditional probability).
- Weisstein, Eric W. "Independent Events." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.