Mathematics and Statistics · Ch 16 — Probability

Conditional Probability

Often we learn that one event has already happened and want the probability of another event in the light of that information. This is conditional probability, written $P(A \mid B)$ and read "the probability of $A$ given $B$ ".

Knowing that $B$ has occurred effectively shrinks the sample space to $B$ alone: only outcomes inside $B$ are now possible, and among those we ask how many also lie in $A$ .

Note

Definition of conditional probability

For events $A$ and $B$ with $P(B) > 0$ ,

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}.$

In counting terms (equally likely outcomes), $P(A \mid B) = \dfrac{n(A \cap B)}{n(B)}$ — the favourable outcomes are those in both $A$ and $B$ , out of the reduced total $n(B)$ .

Symmetrically, when $P(A) > 0$ , $\;P(B \mid A) = \dfrac{P(A \cap B)}{P(A)}$ .

Example. A die is rolled and we are told the result is even ( $B = \{2, 4, 6\}$ ). What is the probability it is greater than $3$ ( $A = \{4, 5, 6\}$ )? The reduced sample space is $B$ with $n(B) = 3$ ; of those, the ones also greater than $3$ are $\{4, 6\}$ , so $n(A \cap B) = 2$ . Hence

$P(A \mid B) = \frac{n(A \cap B)}{n(B)} = \frac{2}{3}.$ …

Definition 11Conditional probability

$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ (with $P(B) > 0$ ): the probability of $A$ given that $B$ has already occurred, computed over the …

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