Explanation
Using probabilities
A probability is a number from 0 (impossible) to 1 (certain). Combining two probabilities depends on how the events relate, so pick the operation that matches your situation.
Which operation to choose
Use Not A for the chance an event does not happen. Use A and B only when the events are independent, meaning one happening does not change the chance of the other. Use A or B when you also know the chance that both happen.
P(not A) = 1 − P(A)
| A | probability of event A |
|---|
P(A and B) = P(A) × P(B)
| A | probability of event A |
|---|---|
| B | probability of event B |
P(A or B) = P(A) + P(B) − P(A and B)
| A | probability of event A |
|---|---|
| B | probability of event B |
The complement of a 1-in-5 chance
| Probability of A | 0.2 |
|---|---|
| Result | 0.8 |
P(not A) = 1 − 0.2 = 0.8
Two independent events both happening
| Probability of A | 0.5 |
|---|---|
| Probability of B | 0.4 |
| Result | 0.2 |
P(A and B) = 0.5 × 0.4 = 0.2
Counting the overlap only once
| Probability of A | 0.5 |
|---|---|
| Probability of B | 0.4 |
| Probability of both A and B | 0.2 |
| Result | 0.7 |
P(A or B) = 0.5 + 0.4 − 0.2 = 0.7
Diagram of this result
Each operation has its own picture: a bar split into A and not A, a unit square whose overlap is A × B, or two overlapping circles labelled with the probability of each part.
Every probability shown in the picture is also stated in text from the same calculation.
Common mistakes
Adding two probabilities counts their overlap twice. Multiplying them gives the chance of both only under independence. A probability of 0.2 means 20%, not 0.2%. The calculator cannot tell whether your events are independent.
Limits and assumptions
All probabilities are decimals from 0 to 1. For A or B, the overlap must lie between max(0, A + B − 1) and min(A, B). No conditional probability, distributions, counting rules or uncertainty in the inputs. Inputs keep up to 30 significant digits and the calculation uses 50; only the displayed answer is rounded.
Sources
OpenStax Introductory Statistics 2e, probability topics: https://openstax.org/details/books/introductory-statistics-2e
Why can an overlap between 0 and 1 still be invalid?
The chance of both cannot exceed the chance of either one, and it must be large enough that A or B does not exceed 1. Both limits are checked.