Multiplication Theorem on Probability

13.3 Multiplication Theorem on Probability

When two events $E$ and $F$ belong to the same sample space $S$, the event that both occur is denoted $E \cap F$, often written $EF$. For instance, drawing two cards one after another, we might want the probability of getting a king and a queen — that is $P(EF)$.

The multiplication theorem computes $P(E \cap F)$ using conditional probabilities. From the definition of conditional probability,

we rearrange to get:

Similarly, since $E \cap F = F \cap E$,

Combining (1) and (2) gives the multiplication rule of probability:

This rule is the foundation for finding probabilities of sequential or simultaneous events when they are not independent.

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Multiplication rule for more than two events

The rule extends to three or more events. For three events $E$, $F$, $G$:

The pattern: multiply the probability of the first event, then the conditional probability of each later event given all the previous ones. For $n$ events:

$P(E_1 \cap E_2 \cap \cdots \cap E_n) = P(E_1) \cdot P(E_2|E_1) \cdot P(E_3|E_1E_2) \cdots P(E_n|E_1E_2\cdots E_{n-1})$ …