S3 Probability — 9MA0 spec coverage

S3.1

Understand and use mutually exclusive and independent events when calculating probabilities; link to discrete and continuous distributions.

Mutually exclusive events cannot occur together, so $P(A\cap B)=0$ and $P(A\cup B)=P(A)+P(B)$ .
Independent events satisfy $P(A\cap B)=P(A)P(B)$ , equivalently $P(A\mid B)=P(A)$ when $P(B)>0$ .
Use $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ for any two events, subtracting the overlap to avoid double-counting.
Events of positive probability cannot be both mutually exclusive and independent; the same rules apply to events defined from discrete or continuous random variables.

Tier 1 · Easy

2 marks

ORIGINAL

Events $A$ and $B$ are mutually exclusive, with $P(A)=0.38$ and $P(B)=0.27$ . Find $P(A\cup B)$ .

Tier 2 · Standard

4 marks

ORIGINAL

Events $A$ and $B$ are independent. Given $P(A)=0.45$ and $P(A\cap B)=0.18$ , find $P(B)$ and $P(A\cup B)$ .

Tier 3 · Hard

5 marks

ORIGINAL

A discrete random variable $X$ has $P(X=0)=0.2$ , $P(X=1)=0.5$ and $P(X=2)=0.3$ . Independently, $Y$ is uniformly distributed on $0\leq y\leq5$ . Let $A$ be the event $X\geq1$ and $B$ the event $Y<2$ . Find $P(A\cap B)$ and $P(A\cup B)$ .

S3.2

Understand and use conditional probability, including the use of tree diagrams, Venn diagrams and two-way tables; understand and use the conditional probability formula P(A|B) = P(A∩B)/P(B).

Conditional probability restricts the sample space: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ for $P(B)>0$ .
On a tree diagram, multiply probabilities along a path and add the probabilities of mutually exclusive paths that satisfy the event.
Without replacement, later branch probabilities change because both the total and the relevant category count may have changed.
In a Venn diagram or two-way table, use the condition as the denominator; reversing $P(A\mid B)$ to $P(B\mid A)$ is a common error.

Tier 1 · Easy

2 marks

ORIGINAL

Of $60$ students, $24$ travel by bus and $15$ of those bus travellers arrive before $8{:}30$ . Find the probability that a randomly chosen bus traveller arrives before $8{:}30$ .

Tier 2 · Standard

4 marks

ORIGINAL

A bag contains $5$ amber counters and $3$ blue counters. Two counters are taken without replacement. Find the probability that exactly one counter of each colour is taken.

Tier 3 · Hard

5 marks

ORIGINAL

For events $A$ and $B$ , $P(A)=0.60$ , $P(B)=0.50$ and $P(A\mid B)=0.70$ . Find $P(A'\mid B')$ .

S3.3

Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions.

A probability model simplifies a real process by specifying possible outcomes and assigning probabilities to them.
State assumptions explicitly, such as independence, constant probabilities, identical trials or equally likely outcomes, and judge them in context.
More realistic dependence or changing probabilities can alter both central probabilities and tail risks, so identify the likely direction of the effect where possible.
Validate a model by comparing its predictions with observed data; a close fit in one sample does not prove that its assumptions are true.

Tier 1 · Easy

2 marks

ORIGINAL

A model assigns probability $1/200$ to each ticket winning a draw. State one assumption behind this model and one reason it might fail.

Tier 2 · Standard

3 marks

ORIGINAL

A factory model treats defects in two items from the same batch as independent, each with probability $0.03$ . It therefore predicts probability $0.03^2$ that both are defective. Explain how batch-to-batch variation is likely to affect this prediction.

Tier 3 · Hard

5 marks

ORIGINAL

A transport model assumes that each of $5$ commuters independently chooses route A with probability $0.4$ . Calculate the modelled probability that all $5$ choose route A. A closure on route B can influence every commuter on the same day. Critique the independence assumption and state the likely effect on the probability just calculated.