<p>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.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an expression for $\mathrm{P}\left( A'\cap B' \right)$ and hence prove that $A'$ and $B'$ are independent events.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">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.)</div><p style="text-align: right;">[2]</p></li></ol>
<p>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 $\mathrm{P}\left( A'\cap B'\cap C' \right)=\frac{1}{10}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find exactly the maximum and minimum possible values of $\mathrm{P}\left( A\cap B \right)$.</div><p style="text-align: right;">[4]</p></li></ol>

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.

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.)

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 $\mathrm{P}\left( A\cap B \right)$.

N2018 P2 Q7

N2018 P2 Q7

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