Probability Calculator
Probability expresses how likely something is, on a scale from 0 to 1. This calculator handles single events and the combinations that follow from them.
Formula
P(event) = favourable outcomes ÷ total outcomes
- Both happening (independent): P(A) × P(B)
- Either happening: P(A) + P(B) − P(A and B)
- At least one: 1 − P(none)
Two coins both heads: ½ × ½ = ¼, or 25%
The Basics
P(event) = Favourable outcomes ÷ Total possible outcomes
Rolling a 4 on a fair die is 1 ÷ 6 = 0.167, or 16.7%.
Probabilities always fall between 0 (impossible) and 1 (certain). The chance of an event not happening is 1 minus the probability that it does, which is often the easier route to an answer.
Combining Events
- Both happening (independent): multiply. Two heads in a row is 0.5 × 0.5 = 0.25.
- Either happening (mutually exclusive): add. Rolling a 1 or a 2 is 1/6 + 1/6 = 1/3.
- Either happening (overlapping): add, then subtract the overlap, or you count it twice.
"And" generally means multiply; "or" generally means add. Because multiplying values below 1 makes them smaller, compound events are always less likely than their parts.
The Gambler's Fallacy
A coin that has landed heads five times running still has a 50% chance of heads next throw. The coin has no memory, and independent events are unaffected by history.
What is true is that five heads in a row was unlikely before you started: 0.55 = 3.1%. Once it has happened, that probability is spent. Confusing the two is the reasoning behind most losing betting systems.
Quick Reference
| Event | Fraction | Percentage | Odds against |
|---|---|---|---|
| Coin lands heads | 1/2 | 50% | 1:1 |
| Die shows a six | 1/6 | 16.7% | 5:1 |
| Two dice total 7 | 6/36 | 16.7% | 5:1 |
| Card is a heart | 13/52 | 25% | 3:1 |
| Card is an ace | 4/52 | 7.7% | 12:1 |
| At least one six in four rolls | — | 51.8% | — |
Frequently Asked Questions
- How do I calculate probability?
- Divide the number of favourable outcomes by the total number of possible outcomes. Rolling a specific number on a die is 1 in 6, or 16.7 per cent.
- When do I add and when do I multiply?
- Multiply for "and" with independent events, add for "or" with mutually exclusive ones. If outcomes overlap, subtract the overlap after adding.
- If a coin lands heads five times, is tails more likely?
- No. Each throw is independent and remains 50 per cent. Expecting a correction is the gambler's fallacy.
- What is the difference between independent and dependent events?
- Independent events do not affect each other, like coin throws. Dependent events do, like drawing cards without replacement, where each draw changes the remaining deck.