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Probability Calculator
Compute single event probability, combined probability of two independent events, or binomial probabilities for a series of trials.
Results
| Occurs — P(A) | — |
|---|---|
| Does Not Occur — P(A') | — |
Results
| Both Occur — P(A ∩ B) | — |
|---|---|
| At Least One Occurs — P(A ∪ B) | — |
| Exactly One Occurs — P(A ⊕ B) | — |
| Neither Occurs | — |
Results
| Exactly k successes — P(X = k) | — |
|---|---|
| At least k successes — P(X ≥ k) | — |
| At most k successes — P(X ≤ k) | — |
| Less than k successes — P(X < k) | — |
| More than k successes — P(X > k) | — |
Understanding Probability Rules
Probability measures the likelihood of an event occurring, represented as a value between 0 (impossible) and 1 (certain), or 0% and 100%.
Single Event: The probability of an event A not occurring is the complement, P(A') = 1 - P(A).
Two Independent Events: When two events A and B do not affect each other:
- Both occur (Intersection): P(A ∩ B) = P(A) * P(B)
- At least one occurs (Union): P(A ∪ B) = P(A) + P(B) - P(A) * P(B)
- Exactly one occurs: P(A ⊕ B) = P(A)(1 - P(B)) + P(B)(1 - P(A))
- Neither occurs: P(neither) = (1 - P(A)) * (1 - P(B))
Binomial Probability Formula
Binomial probability calculates the probability of obtaining exactly k successes in n independent trials, where the probability of success in any single trial is p.
Formula: P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where C(n, k) = n! / (k! * (n - k)!) is the number of combinations of choosing k items from n.
FAQ
What are independent events?
Two events are independent if the occurrence of one has no influence on the probability of the other. For example, rolling a 6 on a die and flipping a heads on a coin are independent events.
Can a probability be negative or greater than 100%?
No. By definition, a probability must be between 0 and 1 (inclusive), or 0% and 100%. Values outside this range are mathematically invalid.