23.6 Conditional probability — Learn

Prerequisites: fractions, Venn diagrams and two-way tables.

Objectives: calculate probability given that an event has happened; identify the restricted total.

Conditional probability means the probability of an event when you know another event has happened. The known event restricts the group you consider. The notation $P(A\mid B)$ means probability of A given B.

Example 1. In a group of 30 students, 18 take the bus. Of those, 10 bring lunch. Among the 12 who do not take the bus, 5 bring lunch. Find the probability that a bus user brings lunch.

The counts can be organised in a two-way table:

The condition is that the student takes the bus, so the restricted total is 18.

Example 2. In a Venn diagram, 15 people are in set $A$ and 6 are in both $A$ and $B$. Find $P(B\mid A)$.

The condition restricts the group to set $A$. The overlap has 6 people, so $P(B\mid A)=\frac6{15}=\frac25$.

Example 3. A bag contains 3 red and 2 blue counters. A counter is drawn and not replaced. Find the probability that the second counter is blue, given that the first was red.

The branches after drawing red show the updated probabilities:

text First draw Second draw Red (3/5) ──────────────┬─ Red (2/4) └─ Blue (2/4) Blue (2/5) ─────────────┬─ Red (3/4) └─ Blue (1/4)

Given red first, 2 blue counters remain among 4. Therefore, $P(\text{second blue}\mid\text{first red})=\frac24=\frac12$.

Watch out. Do not divide by the whole group after a condition is given. “Given bus” makes 18 the denominator in the example, not the full school roll.

Progress check

Of 30 spins, 18 land on red. Of those 18 red spins, 10 land on an odd-numbered sector. Find the probability that a spin lands on an odd sector, given that it is red. State the restricted total.

You should now be able to

• identify the group restricted by the condition;
• calculate conditional probability from a two-way table, a Venn diagram and a tree diagram.