The events $A$, $B$ and $C$ are such that $P(A)=a$, $\mathrm{P}\left( B \right)=b$ an

Question

The events $A$, $B$ and $C$ are such that $P(A)=a$, $\mathrm{P}\left( B \right)=b$ and $\mathrm{P}\left( C \right)=c$. $A$ and $B$ are independent events. $A$ and $C$ are mutually exclusive events. Find an expression for $\mathrm{P}\left( A'\cap B' \right)$ and hence prove that $A'$ and $B'$ are independent events.[2] Find an expression for $\mathrm{P}\left( A'\cap C' \right)$. Draw a Venn diagram to illustrate to case when $A'$ and $C'$ are also mutually exclusive events. (You should not show event $B$on your diagram.)[2] You are now given that $A'$ and $C'$ are not mutually exclusive, $P(A)=\frac{2}{5}$, $\mathrm{P}\left( B\cap C \right)=\frac{1}{5}$ and $\mathrm{P}\left( A'\cap B'\cap C' \right)=\frac{1}{10}$. Find exactly the maximum and minimum possible values of $\mathrm{P}\left( A\cap B \right)$.[4]

Accepted Answer

(i) ${1-a-b+ab}$ (ii) ${1-a-c}$ (iii) ${\frac{2}{15} \leq \mathrm{P}(A \cap B) \leq \frac{1}{3}}$

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