Probability of Combined Events

The AND/OR Rules

Multiplication rule (AND — both events happen): For independent events A and B: $$P(A \text{ and } B) = PA \times PB$$

Addition rule (OR — at least one event happens): For mutually exclusive events: $$P(A \text{ or } B) = PA + PB$$

Tree Diagrams

Tree diagrams show all possible outcomes of two or more events. Each branch is labelled with its probability.

Rules:

• Probabilities on each set of branches from the same point must add to 1.
• To find the probability of a combined outcome: multiply along the branches.
• To find the probability of multiple outcomes satisfying a condition: add the relevant end-branch probabilities.

Example: A bag contains 3 red and 2 blue balls. A ball is drawn, replaced, then drawn again. Find P(both red).

Draw tree:

• First draw: P(R) = 3/5, PB = 2/5
• Second draw (with replacement): same probabilities

$P(RR) = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25}$

Without Replacement (Dependent Events)

When items are not replaced, the second probability changes.

Example: Same bag — now draw without replacement. $P(RR) = \frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10}$

Note: after drawing a red, only 2 red remain out of 4 total.

Finding P(at least one red)

Method 1: P(at least one R) = P(RR) + P(RB) + P(BR)

Method 2 (complement): P(at least one R) = 1 − P(no reds) = 1 − PBB $$= 1 - \frac{2}{5} \times \frac{1}{4} = 1 - \frac{2}{20} = \frac{18}{20} = \frac{9}{10}$$

The complement method is often faster — use it for "at least one" questions.

Conditional Probability

$P(A | B)$ means the probability of A given that B has already happened.

$$P(A | B) = \frac{P(A \text{ and } B)}{PB}$$

From a two-way table or tree diagram, conditional probability is read directly from the relevant subset.

Example: In a class, 15 students study French (F) and 10 study Spanish (S); 6 study both. Find P(F | S). $$P(F | S) = \frac{P(F \cap S)}{P(S)} = \frac{6/25}{10/25} = \frac{6}{10} = \frac{3}{5}$$

WJEC Exam Tips

• Always draw a tree diagram if the question has two or more sequential events.
• Label every branch with a probability and ensure each pair sums to 1.
• With/without replacement — check carefully; it changes the second-row probabilities.
• Use the complement for "at least one" — it saves time.
• Show all working: multiply along branches, add results.