Probability Calculator
Probability measures how likely an event is to happen, expressed as a number between 0 (impossible) and 1 (certain) — or equivalently, between 0% and 100%. Given the individual probabilities of two events, this calculator computes seven related outcomes at once: both events happening together, at least one happening, neither happening, exactly one happening, and the complement of each event on its own.
How the Probability Calculator Works
Enter P(A) and P(B) — the individual probability of each event, as a decimal between 0 and 1 — and the calculator applies the standard rules of probability to derive everything else. The core relationships it computes are:
- P(A ∩ B) — both events happen: P(A) × P(B)
- P(A ∪ B) — at least one happens: P(A) + P(B) − P(A ∩ B)
- P(A′) — A does not happen: 1 − P(A)
- P(neither) — neither event happens: P(A′) × P(B′)
- P(exactly one) — exactly one happens, not both: P(A) + P(B) − 2 × P(A ∩ B)
An Important Assumption Built Into This Calculator: Independence
The multiplication rule this calculator uses — P(A ∩ B) = P(A) × P(B) — is only valid when A and B are independent events, meaning the outcome of one has no effect whatsoever on the probability of the other. Flipping a coin twice, or rolling two separate dice, are independent: what happened on the first flip or roll tells you nothing about the second. Many real situations are not independent, though, and applying this formula to them will give a wrong answer. Drawing two cards from a deck without replacement is a classic example of dependence — after removing one card, the probabilities for the second draw have genuinely changed, because the deck itself is different. If your two events are dependent on each other, you need conditional probability instead — P(A and B) = P(A) × P(B given A) — which requires knowing how A’s occurrence changes B’s probability, information this simple calculator doesn’t ask for or account for.
Union vs. Intersection: The Two Concepts Most Often Confused
The intersection (∩) is the “and” case — both events occur together. The union (∪) is the “or” case — at least one of the two occurs, which includes the possibility of both happening. People frequently calculate a union by simply adding P(A) + P(B), but that double-counts the overlap where both events happen simultaneously — which is exactly why the correct union formula subtracts the intersection back out: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Skipping that subtraction is one of the most common probability mistakes, and it always overestimates the true probability of “at least one.”
Where the Individual Probabilities Actually Come From
This calculator takes P(A) and P(B) as inputs rather than deriving them, which means you need to establish those numbers first — often through basic counting. If an event’s probability comes down to “how many favorable outcomes exist out of how many total outcomes are possible,” the Permutation and Combination Calculator handles that counting step directly: use it to count total possibilities and favorable outcomes separately, divide one by the other to get your P(A) or P(B), then bring those decimal values here to combine them.
Converting a Standardized Score Into a Probability
A common source of a probability input is a normal distribution — for example, “what’s the probability a randomly selected value falls above a certain threshold?” That specific calculation uses a z-score rather than the direct-input approach this calculator takes. The Z-Score Calculator converts a raw value, mean, and standard deviation into exactly that kind of probability, which you can then feed into this calculator as P(A) if you need to combine it with a second, independent event.
From Individual Probabilities to a Full Distribution
This calculator handles exactly two events at a time. Many real problems instead involve counting the probability of a specific number of successes across many repeated independent trials — for example, the probability of getting exactly 6 heads in 10 coin flips. That’s a binomial distribution problem, which builds on the same independence assumption this calculator uses but extends it across many trials at once using combinations to count the arrangements. If you’re working with that kind of repeated-trial scenario, the Permutation and Combination Calculator‘s combination output is the missing piece — it counts how many ways a specific number of successes can occur among the trials, which you then combine with individual event probabilities manually.
Testing Whether a Result Is Statistically Significant
Basic probability rules like the ones this calculator applies are also the foundation underneath formal hypothesis testing — determining whether an observed result is likely due to genuine effect or could plausibly be explained by chance alone. The P-value Calculator builds on these same probability principles, applied specifically to testing a claim against sample data, rather than combining two known event probabilities directly.
Probability in Everyday Random Events
Independent events show up constantly outside formal statistics — rolling a die, flipping a coin, or drawing a random number are all situations where each outcome doesn’t affect the next. If you want to explore these independent events directly rather than just their probabilities in the abstract, the Dice Roller and Random Number Generator let you simulate the actual random draws this calculator’s math describes.
When Your Events Aren’t Independent
If you’re working with a situation involving sequential draws without replacement, medical test results (where a positive test changes the probability of actually having a condition), or any scenario where one event’s outcome genuinely shifts another’s likelihood, you’re in conditional probability territory, which this calculator’s independence-based formulas don’t cover. That kind of problem typically requires working through Bayes’ theorem or a probability tree by hand, using the base probabilities as a starting point rather than a direct plug-in calculation.
Frequently Asked Questions
What is the difference between P(A ∩ B) and P(A ∪ B)?
P(A ∩ B) is the probability that both events A and B happen together. P(A ∪ B) is the probability that at least one of them happens, which includes the case where both occur. The union is always greater than or equal to the intersection.
Can I use this calculator if my two events aren’t independent?
Not accurately. This calculator assumes A and B are independent, meaning one event’s outcome doesn’t affect the other’s probability. If your events are dependent — like drawing cards without replacement — you need conditional probability instead, which accounts for how the first event changes the second event’s likelihood.
Why do you subtract the intersection when calculating a union?
Because simply adding P(A) + P(B) counts the overlap — the outcomes where both events happen — twice. Subtracting P(A ∩ B) once corrects for that double-counting, giving the true probability that at least one event occurs.
What does P(A′) mean?
P(A′), the complement of A, is the probability that event A does not happen. It’s always calculated as 1 minus P(A), since an event either happens or it doesn’t — there’s no other possibility.
How do I find the probability of two dependent events happening together?
Use conditional probability: P(A and B) = P(A) × P(B given A), where P(B given A) is the probability of B happening specifically after A has already occurred. This requires knowing how A’s occurrence changes the likelihood of B, which independent-event formulas like the ones in this calculator don’t account for.