How this probability calculator works
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For a single event with equally likely outcomes, the probability is simply the number of favorable outcomes divided by the total number of possible outcomes: P = favorable ÷ total. The same value can be shown three ways — as a fraction, as a decimal, and as a percentage — and they all describe the same chance.
Every event also has a complement, written P(not A), which is the chance the event does not happen. Because one outcome must occur, the two always add to 1, so P(not A) = 1 − P(A). When you combine two independent events — events where one outcome does not affect the other — the rules are: P(A and B) = P(A) × P(B), P(A or B) = P(A) + P(B) − P(A) × P(B), and P(neither) = (1 − P(A)) × (1 − P(B)).
Common probability rules
| Quantity | Formula | Applies to |
|---|---|---|
| Single event P(A) | favorable ÷ total | Equally likely outcomes |
| Complement P(not A) | 1 − P(A) | Any event |
| P(A and B) | P(A) × P(B) | Independent events |
| P(A or B) | P(A) + P(B) − P(A) × P(B) | Independent events |
| P(neither) | (1 − P(A)) × (1 − P(B)) | Independent events |
Reference note: the AND, OR and neither rules above assume the two events are independent. If one event changes the probability of the other, the events are dependent and require conditional probability, which is not covered here.
Frequently asked questions
- How do I calculate probability?
- For a single event with equally likely outcomes, divide the number of favorable outcomes by the total number of possible outcomes: P = favorable ÷ total. For example, rolling a 3 on a six-sided die is 1 ÷ 6 ≈ 0.167, or about 16.7%. The result is always between 0 and 1.
- What is P(A and B)?
- P(A and B) is the probability that both events happen. For two independent events it is the product of their probabilities: P(A and B) = P(A) × P(B). For example, two fair coins both landing heads is 0.5 × 0.5 = 0.25, or 25%.
- What is P(A or B)?
- P(A or B) is the probability that at least one of the events happens. For independent events it is P(A or B) = P(A) + P(B) − P(A) × P(B). Subtracting the overlap prevents counting the case where both occur twice.
- What are independent events?
- Two events are independent when the outcome of one does not change the probability of the other — for example separate coin flips or dice rolls. Only for independent events does P(A and B) equal P(A) × P(B). If events influence each other they are dependent and need conditional probability.
- What is the complement rule?
- The complement of an event A is everything in which A does not happen, written P(not A). Because some outcome must occur, P(not A) = 1 − P(A). For example, if the probability of rain is 0.3, the probability of no rain is 1 − 0.3 = 0.7.
- Can probability exceed 1?
- No. A probability is always between 0 and 1 inclusive, or 0% to 100%. A value of 0 means the event cannot happen and 1 means it is certain. If a calculation gives a number above 1 or below 0, an input is wrong — for example more favorable outcomes than total outcomes.