The events $A$ and $B$ are such that $\mathrm{P}\left( A \right)=0.3$, $\mathrm{P}\left( B' \right)=

Question

The events $A$ and $B$ are such that $\mathrm{P}\left( A \right)=0.3$, $\mathrm{P}\left( B' \right)=0.85$ and $\mathrm{P}\left( \left. B \right|A \right)=0.15$. Find $\mathrm{P}\left( A'\cap B' \right)$ and hence prove that the events $A'$ and $B'$ are independent events. [4] A third event $C$ is such that $\mathrm{P}\left( C \right)=0.45$ and $\mathrm{P}\left( A\cap B\cap C \right)=0.03$. It is also given that events $B$ and $C$ are independent, and that events $A$ and $C$ are not independent. Find the range of possible values of $\mathrm{P}\left( A'\cap B'\cap C' \right)$. [5]

Accepted Answer

$A'$ and $B'$ are independent; $0.2125 \leq \text{P}(A' \cap B' \cap C') \leq 0.4675$, $\text{P}(A' \cap B' \cap C') \neq 0.3125$

2022 SAJC J2 CT Q9

2022 SAJC J2 CT Q9

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