AI Tutor
If you flip a coin and get heads ten times in a row, is the eleventh flip more likely to land on tails?
Intuitively, it feels like tails is "due," but the coin has no memory — the chance of heads remains exactly 0.50.5 on every single flip.

What Makes Events Independent?

Two events AA and BB are independent if the outcome of one event does not change the likelihood of the other event occurring.
In probability notation, we write this as P(A∣B)=P(A)P(A \mid B) = P(A), where the conditional probability P(A∣B)P(A \mid B) represents the probability of AA occurring given that BB has already happened.
In 1718, French mathematician Abraham de Moivre formalized this definition in his book The Doctrine of Chances while analyzing sequential probability in casino games.
How can we prove the classic formula for computing the joint probability of two independent events?

The Multiplication Rule

The general multiplication rule states that the probability of both events happening is P(A∩B)=P(A)⋅P(B∣A)P(A \cap B) = P(A) \cdot P(B \mid A), where the symbol ∩\cap means intersection (both events occurring together).
If event AA has no influence on event BB, then P(B∣A)=P(B)P(B \mid A) = P(B), which simplifies the formula directly to P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B).
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