[Leave your answers in fraction] The events $A$ and $B$ are such that $\mathrm{P}\left( A|B \right)=

Question

[Leave your answers in fraction] The events $A$ and $B$ are such that $\mathrm{P}\left( A|B \right)=\frac{7}{10}$, $\mathrm{P}\left( B|A \right)=\frac{4}{15}$ and $\mathrm{P}\left( A\cup B \right)=\frac{3}{5}$. Find the exact values of $\mathrm{P}\left( A\cap B \right)$, [3] $\mathrm{P}\left( A'\cap B \right)$. [2] Find a third event $C$, it is given that $\mathrm{P}\left( C \right)=\frac{3}{10}$ and that $A$ and $C$ are independent. Find $\mathrm{P}\left( A'\cap C \right)$. [2] Hence state an inequality satisfied by $\mathrm{P}\left( A'\cap B\cap C \right)$. [1]

Accepted Answer

(i) $$\text{P}(A \cap B) = \frac{28}{195}$$ (ii) $$\frac{4}{65}$$ (iii) $$\frac{9}{65}$$ (iv) $$0 \leq \text{P}(A' \cap B \cap C) \leq \frac{4}{65}$$

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