The Venn diagram shows the probabilities associated with three events, $A$, $B$ and $C$.
(a) Show that the events $A$ and $B$ are independent. [2]
(b) Find $\mathrm{P}\big((B\cup C)'\big)$. [1]
(c) Write down two of the events that are mutually exclusive. Give a reason for your answer. [1]
- (a) finds $\mathrm{P}(A)=0.26+0.14=0.4$ and $\mathrm{P}(B)=0.14+0.16+0.05=0.35$ and attempts $\mathrm{P}(A)\times\mathrm{P}(B)$ (or attempts $\mathrm{P}(A\mid B)=\frac{0.14}{0.35}$ to compare with $\mathrm{P}(A)$, or $\mathrm{P}(B\mid A)=\frac{0.14}{0.4}$ to compare with $\mathrm{P}(B)$)M1
- (a) $0.4\times0.35=0.14=\mathrm{P}(A\cap B)$ (or $\mathrm{P}(A\mid B)=0.4=\mathrm{P}(A)$, or $\mathrm{P}(B\mid A)=0.35=\mathrm{P}(B)$), so $A$ and $B$ are independent (conclusion required)A1
- (b) $0.26+0.24=0.5$ (accept $1-0.50$ or equivalent)B1
- (c) $A$ and $C$, with a reason, e.g. $\mathrm{P}(A\cap C)=0$ / the circles do not overlap / they have no outcomes in commonB1
- Answer: (a) $0.4\times0.35=0.14=\mathrm{P}(A\cap B)$; (b) 0.5; (c) $A$ and $C$
- Examiner tip: To prove independence you must show $\mathrm{P}(A)\times\mathrm{P}(B)=\mathrm{P}(A\cap B)$ with the numbers and write a conclusion.