Probability Calculator
The chance of both, either, neither or exactly one of two independent events.
Results
What this tool does
Two probabilities are easy to hold in your head; what they imply together is not. Two events that are each 50% likely happen together only a quarter of the time, and the chance that at least one occurs is 75%, not 100% — which is the intuition most people have to correct once and then never forget. Enter each probability as a percentage and the four combinations come out at once, along with the odds form that betting and clinical papers use.
Formula
P(A and B) = P(A) P(B) ; P(A or B) = P(A) + P(B) − P(A) P(B)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
pa | Probability of A | % |
pb | Probability of B | % |
BO | Both happen | % |
EI | At least one happens | % |
NE | Neither happens | % |
OA | A happens and B does not | % |
EX | Exactly one happens | % |
NA | A does not happen | % |
OD | Odds against | — |
Worked example
- Probability of A40 %
- Probability of B25 %
- Both happen10.0000 %
- At least one happens55.0000 %
- Neither happens45.0000 %
- A happens and B does not30.0000 %
- Exactly one happens45.0000 %
- A does not happen60.0000 %
- Odds against1.5000
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
Frequently asked questions
What does independent mean here?
That knowing one event happened tells you nothing about the other. Two coin tosses are independent; rain today and rain tomorrow are not. The whole page assumes independence, and when that assumption is wrong the answers can be badly wrong in either direction — this is exactly the mistake that made bundles of mortgages look far safer than they were.