How the Probability Calculator Works
Probability measures how likely something is. It is a number from 0 (it cannot happen) to 1 (it will certainly happen). A fair coin landing on heads has a probability of 0.5, which is 50%. The Probability Calculator works out the chance that two events both happen, and the chance that at least one of them happens.
What you enter
- Events: choose Independent or Mutually exclusive (see below).
- P(A): the probability of event A, from 0 to 1.
- P(B): the probability of event B, from 0 to 1.
To use a percentage, divide by 100: 30% is 0.3.
What you get
- P(A and B) (the intersection): both events happen.
- P(A or B) (the union): A happens, or B happens, or both.
Each result is shown as a decimal and as a percentage.
The two kinds of events
| Kind | Meaning | Formulas |
|---|---|---|
| Independent | One event does not change the chance of the other | P(A and B) = P(A) x P(B) P(A or B) = P(A) + P(B) - P(A and B) |
| Mutually exclusive | They cannot both happen at once | P(A and B) = 0 P(A or B) = P(A) + P(B) |
- Kind: Independent
Meaning: One event does not change the chance of the other
Formulas: P(A and B) = P(A) x P(B)
P(A or B) = P(A) + P(B) - P(A and B) - Kind: Mutually exclusive
Meaning: They cannot both happen at once
Formulas: P(A and B) = 0
P(A or B) = P(A) + P(B)
For mutually exclusive events, P(A) + P(B) cannot be more than 1, and the calculator warns you if it is.
A worked example
The default is two independent events with P(A) = 0.5 and P(B) = 0.4.
| Step | Value |
|---|---|
| P(A and B) = 0.5 x 0.4 | 0.2 (20%) |
| P(A or B) = 0.5 + 0.4 - 0.2 | 0.7 (70%) |
More examples
| Situation | Type | Result |
|---|---|---|
| Rain 0.3, a traffic jam 0.2: both? | Independent | 0.06 (6%) |
| Rain 0.3, a traffic jam 0.2: either? | Independent | 0.44 (44%) |
| Roll a 1 (1/6) or a 2 (1/6) on one die | Exclusive | 0.333333 (33.33%) |
| Two coin flips, both heads (0.5 and 0.5) | Independent | 0.25 (25%) |
- Situation: Rain 0.3, a traffic jam 0.2: both?
Type: Independent
Result: 0.06 (6%) - Situation: Rain 0.3, a traffic jam 0.2: either?
Type: Independent
Result: 0.44 (44%) - Situation: Roll a 1 (1/6) or a 2 (1/6) on one die
Type: Exclusive
Result: 0.333333 (33.33%) - Situation: Two coin flips, both heads (0.5 and 0.5)
Type: Independent
Result: 0.25 (25%)
Chart: the default example
Each bar is a probability. The scale is the percentage:
P(A) ████████████████████ 50% P(B) ████████████████ 40% P(A and B) ████████ 20% P(A or B) ████████████████████████████ 70%
Chart: several heads in a row
The chance of getting heads every time on a fair coin halves with each flip:
1 flip ████████████████████████████████████████ 0.5 2 flips ████████████████████ 0.25 3 flips ██████████ 0.125 4 flips █████ 0.0625 5 flips ███ 0.03125
Tips
- Check independence. Two coin flips are independent. Drawing two cards from a deck without putting the first back is not.
- "Or" is not just addition. For independent events, subtract the overlap, or you count it twice.
- Probabilities must be from 0 to 1. A value like 50 (for 50%) is rejected.
- Results are rounded to 6 decimal places.
This tool gives an estimate for general information only. Real-life events are often not perfectly independent.