Probability CalculatorFree online math calculator — instant results, no sign-up
Combined probabilities of two independent events.
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About the Probability Calculator
This calculator computes the combined probabilities of two independent events, A and B. It determines the likelihood of both events occurring, either event occurring, neither occurring, exactly one occurring, and the complement of event A. Students, researchers, and analysts use it to quickly evaluate joint scenarios under the assumption of statistical independence.
- P(A)
- Probability of event A occurring, expressed as a decimal between 0 and 1.
- P(B)
- Probability of event B occurring, expressed as a decimal between 0 and 1.
Formula and method
P(A and B) = P(A) × P(B) P(A or B) = P(A) + P(B) − [P(A) × P(B)] P(not A) = 1 − P(A) P(exactly one) = P(A) × [1 − P(B)] + P(B) × [1 − P(A)] P(neither) = [1 − P(A)] × [1 − P(B)]
- P(A)Probability that event A occurs
- P(B)Probability that event B occurs
- P(A and B)Joint probability of both independent events occurring simultaneously
- P(A or B)Probability of at least one of the two events occurring (union)
- P(not A)Complementary probability that event A does not occur
- P(exactly one)Probability that either A occurs without B, or B occurs without A
- P(neither)Probability that neither event A nor event B occurs
The calculator treats event A and event B as statistically independent, meaning the outcome of one has no influence on the other. It evaluates the standard algebraic formulas for intersection, union, complement, symmetric difference, and joint non-occurrence, rounding each resulting value to 6 decimal places.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- P(A)0.5
- P(B)0.4
Results
- P(A and B)0.2
- P(A or B)0.7
- P(not A)0.5
- P(exactly one)0.5
- P(neither)0.3
How to use the Probability Calculator
- 1Enter the individual probability for event A as a decimal from 0 to 1.
- 2Enter the individual probability for event B as a decimal from 0 to 1.
- 3Multiply P(A) and P(B) to find the joint probability P(A and B).
- 4Calculate the union, complements, exclusive outcomes, and neither-event probabilities using standard independence equations.
Reading your result
Values are returned as decimals between 0 (impossible) and 1 (certain). For instance, P(A or B) gives the chance of at least one event happening, while P(neither) shows the chance that both fail to occur. You can multiply any resulting decimal by 100 to view it as a percentage.
Limitations
The formulas strictly assume that events A and B are independent. If the occurrence of event A changes the likelihood of event B (dependent events), or if the events cannot happen at the same time (mutually exclusive events), these formulas will yield incorrect results.
Common questions
What does it mean for two events to be independent?+
Two events are independent if the occurrence of one does not affect the probability of the other, such as flipping a coin twice or rolling two separate dice.
Can P(A or B) ever exceed 1.0?+
No. Because the formula subtracts the joint probability P(A and B) from the sum of P(A) and P(B), the union will never exceed 1.0 for valid input probabilities.
How can I convert percentage chances into the required input format?+
Divide the percentage value by 100 before entering it; for example, a 25% chance should be entered as 0.25.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.