Conditional probability and independence

Independent events

Two events are independent if the outcome of one does not affect the probability of the other.

For independent events: P(A and B) = PA × PB.

Example: Flipping a coin twice. P(head then head) = 1/2 × 1/2 = 1/4.

Dependent events

The outcome of one event changes the probability of the next. Most commonly seen in problems where items are picked without replacement.

Conditional probability notation

P(B | A) = the probability of B given that A has happened.

For independent events: P(B | A) = PB. For dependent events: P(B | A) ≠ PB.

Conditional probability formula

P(A and B) = PA × P(B | A).

Rearranged: P(B | A) = P(A and B) ÷ PA.

Tree diagrams (without replacement)

CCEA Higher commonly tests this. Critical: the second-branch probabilities depend on what happened on the first.

Example: Bag has 4 red and 6 blue balls. Two are drawn without replacement.

• First red: 4/10. Then second red: 3/9 (one red removed from total of 9).
• First red: 4/10. Then second blue: 6/9.
• P(both red) = 4/10 × 3/9 = 12/90 = 2/15.

Two-way tables and conditional probability

Given a two-way table, P(A | B) = (cell where both occur) ÷ (row/column total for B).

Example: a table shows 50 students. 30 study Maths; of those, 18 also study French. P(French | Maths) = 18/30 = 3/5.

Common CCEA exam tip

For without-replacement, always re-check the denominator — total reduces by 1. A wrong denominator costs the M1.