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).
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Analogy
Flipping a coin in London does not change a die roll in Delhi.
Example
P(head) = 1/2P(six) = 1/6P(head and six) = 1/2 x 1/6 = 1/12Three 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.