Conditional probability

Conditional probability P(A | B) from two-way tables, Venn diagrams and the formula P(A ∩ B) ÷ P(B).

A smaller sample space

$P(A \mid B)$ means "the probability of $A$, given $B$ has happened". Only the outcomes in $B$ count.

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

Example 1●●

Of $40$ students, $15$ study French, $20$ study Japanese and $6$ study both. A Japanese student is chosen. Find the probability they also study French.

Restrict to the $20$ Japanese students; $6$ also study French.

$P(F \mid J) = \dfrac{6}{20} = \dfrac{3}{10}$

Example 2●

A die shows an even number. Find the probability it is a $6$.

Given even: $\{2, 4, 6\}$. $P = \dfrac{1}{3}$.

Flashcards

$P(A mid B)$

$\dfrac{P(A \cap B)}{P(B)}$

Given that...

Shrink the sample space.

$P(6 mid ext{even})$

$\tfrac{1}{3}$

Practice

$40$ students: $15$ French, $20$ Japanese, $6$ both. $P(F \mid J)$?
Same students. $P(J \mid F)$?
Die shows even. $P(6)$?
$P(A \cap B) = 0.12$, $P(B) = 0.4$. Find $P(A \mid B)$.
A card drawn from a standard deck is a heart. Find the probability it is a jack, queen or king.
Same $40$ students. $P(\text{neither})$?
Two coins are tossed and at least one is a head. Find $P(\text{exactly one head})$.
$P(A) = 0.6$, $P(A \cap B) = 0.3$. Find $P(B \mid A)$.