Conditional Probability Calculator
Enter the probabilities of events A and B along with their intersection to instantly compute conditional probabilities and related measures.
P(A | B)
50.00%
Probability of A given B
P(B | A)
37.50%
Probability of B given A
P(A ∪ B)
55.00%
Probability of at least one event
| Region | Probability |
|---|---|
| Only A | 25.00% |
| Only B | 15.00% |
| A ∩ B | 15.00% |
| Neither | 45.00% |
How to Use This Calculator
- Enter P(A), P(B), and P(A ∩ B) using decimal values between 0 and 1.
- Review the computed conditional probabilities P(A | B) and P(B | A).
- Check the union and exclusive regions to understand how the probabilities distribute across the Venn diagram.
- Use the error messages to correct inconsistent inputs such as impossible intersections.
Formula
P(A | B) = P(A ∩ B) / P(B)
P(B | A) = P(A ∩ B) / P(A)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(neither) = 1 − P(A ∪ B)
Conditional probabilities require dividing by the probability of the given event. If P(B) or P(A) equals zero, the corresponding conditional probability is undefined.
Full Description
Conditional probability quantifies how the likelihood of one event changes when another event is known to occur. This calculator helps you keep track of the relationships between P(A), P(B), their intersection, and derived quantities such as unions and complements. It automatically checks for consistency using the inclusion-exclusion principle.
Understanding these relationships is essential for risk analysis, medical testing, and statistical inference. Once you master conditional probability, many paradoxes — such as Bayes' theorem scenarios — become more intuitive.
Frequently Asked Questions
Why can the calculator display a dash?
When P(A) or P(B) equals zero, the corresponding conditional probability is undefined. In that case the calculator displays “—” to indicate the division cannot be performed.
What if the intersection is impossible?
The inputs violate probability axioms if P(A ∩ B) exceeds either P(A) or P(B), or if it is smaller than P(A) + P(B) − 1. The calculator alerts you so you can fix the numbers.
Can I use percentages instead of decimals?
Convert percentages to decimals before entering them. For example, 45% becomes 0.45. This keeps the formulas straightforward and avoids ambiguity.
Does this handle independent events?
Yes. Independent events satisfy P(A ∩ B) = P(A) · P(B). Entering consistent values will produce equal conditional probabilities with the original event probabilities.