# Independent events

Events are independent if one happening does not change the chance of the other. Then P(A and B) = P(A) x P(B).

Statistics (university) · Part 02: Probability · Card 4 of 8 · Last updated 2026-10-19

## Analogy

Flipping a coin in London does not change a die roll in Delhi.

## Example

**Coin and die**

P(head) = 1/2
P(six) = 1/6
P(head and six) = 1/2 x 1/6 = 1/12
Three heads in a row:
1/2 x 1/2 x 1/2 = 1/8

## In practice

Separate coin flips, rolls of different dice and unrelated machines are usually modelled as independent.

## Practice

### Flashcards

- **What is P(A and B) for independent events?** P(A) x P(B).
- **Can mutually exclusive events with positive probability be independent?** No. If one happens the other cannot, so they affect each other.

## Questions people ask

### If I flip heads five times, is tails more likely next?

No. Each flip of a fair coin is independent, so the chance of heads stays 0.5 whatever happened before. Believing tails is due is the gambler's fallacy. The coin has no memory, and the long-run balance comes from averaging, not correction.

### How do you test whether two events are independent?

Check whether P(A and B) equals P(A) x P(B). If it does, they are independent. Equivalently, P(A given B) should equal P(A). Real data rarely matches exactly, because samples are noisy, so with data you use a statistical test such as chi-square to judge whether the gap is just chance.

## Related concepts

- [Mutually exclusive events](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-2/statistics-university-part-2-concept-3)
- [Addition rule](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-2/statistics-university-part-2-concept-2)

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