Probability of combined events, mutually exclusive and independent events, conditional probability, discrete random variables with their expectation and variance, and the binomial distribution as a model.

What this dot point is asking

CCEA wants you to calculate probabilities of combined events, distinguish mutually exclusive from independent events, use conditional probability, work with discrete random variables (their distribution, expectation and variance), and recognise and apply the binomial distribution as a model.

The answer

Probability of combined events

Independence and conditional probability

Discrete random variables

The binomial distribution

Examples in context

Example 1. Reliability of a delivery service. If each parcel arrives on time with probability $p$ independently, the number of on-time deliveries in a batch of $n$ follows $B(n, p)$. The expected number $np$ tells a manager the typical service level, and the tail probabilities flag unusually bad days.

Example 2. Medical screening. Conditional probability underlies test interpretation: the chance a patient actually has a condition given a positive test, $P(\text{ill} \mid +)$, depends on how common the condition is. Confusing this with $P(+ \mid \text{ill})$ is a classic real-world error the conditional formula corrects.

Try this

Q1. Events $A$ and $B$ are mutually exclusive with $P(A) = 0.3$, $P(B) = 0.4$. Find $P(A \cup B)$. [1 mark]

• Cue. $0.3 + 0.4 = 0.7$ (since $P(A \cap B) = 0$).

Q2. For $X \sim B(10, 0.3)$, write down $E(X)$. [1 mark]

• Cue. $E(X) = np = 10(0.3) = 3$.

Q3. A discrete variable has $P(X = 0) = 0.6$, $P(X = 1) = 0.4$. Find $E(X)$. [1 mark]

• Cue. $0(0.6) + 1(0.4) = 0.4$.