Probability calculator
Work through intersection, union, and conditional probability for two events.
What this calculator covers
Use this calculator for two-event probability work when you want the overlap, union, and conditional relationship shown together.
The independent-versus-dependent toggle keeps the assumptions explicit so you can see whether the overlap comes from multiplication or from a supplied shared probability.
Frequently asked questions
- What is the difference between independent and dependent events?
- Two events are independent when knowing one occurred gives no information about whether the other occurred. They are dependent when there is a real-world connection — for example, drawing two cards without replacement, where the first draw changes the odds for the second.
- How does the union probability relate to the individual event probabilities?
- The union P(A ∪ B) is the probability that at least one of the two events occurs. It equals P(A) + P(B) minus the overlap P(A ∩ B), because simply adding the two probabilities would count the shared region twice.
- What does conditional probability P(A|B) mean?
- P(A|B) is the probability that event A occurs given that event B has already occurred. It narrows the sample space to only the outcomes where B happened, then asks what fraction of those also include A.
- Why must I enter P(B) greater than zero?
- Conditional probability divides by P(B), so a zero denominator is undefined. If event B is impossible it cannot serve as the condition for any other event.
Tool
Run the calculation
Result
RESULT · P(A ∪ B)
â„–161
Primary result
58%
With P(A) = 40% and P(B) = 30%, the union probability P(A ∪ B) is 58% and the conditional probability P(A|B) is 40%.
- P(A)
- 40%
- P(B)
- 30%
- P(A ∩ B)
- 12%
- P(A ∪ B)
- 58%
- P(A|B)
- 40%
Step-by-step solution
- 1.Because the events are treated as independent, P(A ∩ B) = P(A) × P(B) = 12%.
- 2.Apply inclusion-exclusion: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 58%.
- 3.Compute the conditional probability from the overlap: P(A|B) = P(A ∩ B) / P(B) = 40%.
Walkthrough
Visual walkthrough
Probability work here centers on overlap, then branches into union and conditional views of the same events.
01
Establish the overlap
P(A ∩ B) = 12%
Independent events multiply to get their shared probability.
02
Build the union
40% + 30% - 12%
Inclusion-exclusion adds the two event probabilities, then removes the overlap once so it is not double-counted.
03
Read the conditional view
Conditional probability narrows the denominator to event B and asks what share of B also lands in A.
40% conditional · 58% union