<p>For events $A$ and $B$, it is given that $P(A\cup B)=0.8$ and $P(\left. A \right|B)=0.5$. It is also given that events $A$ and $B$ are independent.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{P(}B\mathrm{)}$.</span></div><p style="text-align: right;">[2]</p></li></ol>
<p>A third event $C$ is such that events $A$ and $C$ are independent and events $B$ and $C$ are independent.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Given also that $\mathrm{P}(C)=0.5$, find exactly the maximum and minimum possible values of $\mathrm{P}(A\cap B\cap C)$.</span></div><p style="text-align: right;">[3]</p></li></ol>

For events $A$ and $B$, it is given that $P(A\cup B)=0.8$ and $P(\left. A \right|B)=0.5$. It is also given that events $A$ and $B$ are independent.

A third event $C$ is such that events $A$ and $C$ are independent and events $B$ and $C$ are independent.

Given also that $\mathrm{P}(C)=0.5$, find exactly the maximum and minimum possible values of $\mathrm{P}(A\cap B\cap C)$.

Probability - General

Probability

International Baccalaureate Higher Level Mathematics

IB HL Math

2020 ACJC P2 Q5

$\mathrm{P}(A|B) = 0.5$. Since $A$ and $B$ independent: $\mathrm{P}(A) = \mathrm{P}(A|B) = 0.5$

$\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A)\mathrm{P}(B)$

$0.8 = 0.5 + \mathrm{P}(B) - 0.5\,\mathrm{P}(B)$

$0.3 = 0.5\,\mathrm{P}(B) \Rightarrow \mathrm{P}(B) = 0.6$

$\mathrm{P}(A \cap C) = 0.5 \times 0.5 = 0.25$, $\mathrm{P}(B \cap C) = 0.6 \times 0.5 = 0.3$, $\mathrm{P}(A \cap B) = 0.3$

$C$ only region: $0.5 - 0.25 - 0.3 + x = x - 0.05 \geq 0 \Rightarrow x \geq 0.05$

$A \cap C$ only: $0.25 - x \geq 0 \Rightarrow x \leq 0.25$

Outside region: $1 - (0.75 + x) = 0.25 - x \geq 0 \Rightarrow x \leq 0.25$

$\therefore 0.05 \leq \mathrm{P}(A \cap B \cap C) \leq 0.25$

2020 ACJC P2 Q5

2020 ACJC P2 Q5

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