Probability Calculator
Find the chance of a single event, of two independent events both or either happening, or of at least one success across several trials.
1 favorable out of 6 equally likely outcomes
Probability is favorable outcomes ÷ total equally likely outcomes, so rolling one specific number on a die is 1 ÷ 6 ≈ 0.1667 = 16.7%. For two independent events both happening, multiply: P(A and B) = P(A) × P(B). Two coins both landing heads is ½ × ½ = 0.25 = 25%.
What probability measures
Probability is a number from 0 to 1 that scores how likely an event is: 0 means it never happens, 1 means it always happens, and 0.5 is an even chance. When every outcome is equally likely, you get it by counting — divide the outcomes you care about by the total number of outcomes. The same value can be written three ways: a fraction (1⁄6), a decimal (0.1667), or a percent (16.67%). Combining events is where it gets interesting: whether you multiply or add depends on whether you want both events, either event, or none of them.
valid only when every outcome is equally likely
Worked example
What is the chance of getting at least one six when you roll a fair die four times?
- 1 Find the single-event probability. One six out of six faces: P(six) = 1 ÷ 6 ≈ 0.1667.
- 2 Flip to the complement. The chance of not rolling a six on one roll is 1 − 1⁄6 = 5⁄6 ≈ 0.8333.
- 3 Multiply across the independent trials. No six in all four rolls: (5⁄6)⁴ ≈ 0.4823.
- 4 Subtract from 1. P(at least one six) = 1 − 0.4823 = 0.5177 ≈ 51.8%.
Combining events
Formulas for independent events A and B with probabilities P(A) and P(B).
| You want | Formula | Reads as |
|---|---|---|
| Both (A and B) | P(A) × P(B) | A happens and B also happens |
| Either (A or B) | P(A) + P(B) − P(A) × P(B) | at least one of A or B happens |
| Neither | (1 − P(A)) × (1 − P(B)) | A fails and B fails |
| At least one in n trials | 1 − (1 − p)ⁿ | success on one or more of n attempts |
Independent vs. mutually exclusive — and the complement trick
Independent events don’t affect each other, so their joint probability multiplies: two coin flips, or rolls of separate dice. Mutually exclusive events can’t both occur in a single trial — rolling a 2 and a 5 on one die — so for “either” you simply add, because the overlap term P(A and B) is zero. Mixing these up is the most common probability mistake: only add raw probabilities when the events truly cannot co-occur.
The complement trick turns awkward “at least one” questions into easy ones. Instead of adding up the chances of exactly one, exactly two, and so on, compute the one case you’re trying to avoid — zero successes — and subtract from 1. That is why P(at least one) = 1 − (1 − p)ⁿ: (1 − p)ⁿ is the chance every trial fails, and everything else is “at least one.”