Probability of events, Venn diagrams and set notation, the addition rule, mutually exclusive and independent events, and tree diagrams.

What this dot point is asking

WJEC wants you to calculate probabilities for single and combined events, to use Venn diagrams and set notation, to apply the addition rule, to recognise mutually exclusive and independent events, and to use tree diagrams for sequences of events with and without replacement. This sets up the binomial distribution and hypothesis testing that follow, and conditional probability in A2.

The answer

Probability of events

The probability of an event $A$ is $P(A) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}$ for equally likely outcomes, with $0 \le P(A) \le 1$. The complement rule gives $P(\text{not } A) = 1 - P(A)$, which is often the quickest route to "at least one" probabilities.

Venn diagrams and set notation

A Venn diagram shows events as overlapping regions. The notation: $A \cap B$ is the intersection (both), $A \cup B$ is the union (either or both), and $A'$ is the complement (not $A$).

The addition rule and mutual exclusivity

You subtract the intersection because outcomes in the overlap would otherwise be counted twice.

Independence and tree diagrams

Examples in context

Example 1. At least one success. A test is passed with probability $0.8$ each attempt, independently. The probability of passing in at least one of two attempts is $1 - P(\text{fail both}) = 1 - 0.2^2 = 1 - 0.04 = 0.96$. The complement turns a messy "at least one" into a single subtraction.

Example 2. Reading a Venn diagram. Of 50 students, 30 study French, 25 study German and 12 study both. Then $P(\text{French or German}) = \dfrac{30 + 25 - 12}{50} = \dfrac{43}{50}$, and $\dfrac{7}{50}$ study neither. The overlap of $12$ is subtracted to avoid counting the bilingual students twice.

Try this

Q1. $P(A) = 0.6$ and $P(A')$ is its complement. Find $P(A')$. [1 mark]

• Cue. $P(A') = 1 - 0.6 = 0.4$.

Q2. Two fair coins are tossed. Find the probability of exactly one head. [2 marks]

• Cue. Outcomes HT and TH each have probability $\tfrac{1}{4}$, total $\tfrac{1}{2}$.

Q3. Events $A$ and $B$ are mutually exclusive with $P(A) = 0.3$, $P(B) = 0.45$. Find $P(A \cup B)$. [2 marks]

• Cue. No overlap, so $P(A \cup B) = 0.3 + 0.45 = 0.75$.