Probability: Conditional; independence — DHS 2025 H2 Math Prelim Paper 2
Dunman High School (DHS)2025 PrelimPaper 2●●● Challenging7 marks
What this question tests
Conditional; independence.
Question
The events \(A\), \(B\) and \(C\) are such that \(\mathrm{P}(A) = 0.8\), \(\mathrm{P}(B) = 0.2\) and \(\mathrm{P}(C) = 0.6\). It is also known that \(\mathrm{P}(A \cap B) = \mathrm{P}(B)\) and \(\mathrm{P}(B \cap C) = 0.1\).
- Find exactly the maximum and minimum possible values of
- \(\mathrm{P}(A \cap B' \cap C)\),
- \(\mathrm{P}\!\left(A \cap B' \cap C \mid A \cup C\right)\).
- It is given further that \(A\) and \(C\) are independent. Find the value of \(\mathrm{P}(A \cap B' \cap C)\).
Show full worked solution▾
(a)(i)
(a)(ii)
\[\mathrm{P}\!\left(A \cap B' \cap C \mid A \cup C\right) = \frac{\mathrm{P}\!\bigl((A \cap B' \cap C) \cap (A \cup C)\bigr)}{\mathrm{P}(A \cup C)} = \frac{\mathrm{P}(A \cap B' \cap C)}{\mathrm{P}(A \cup C)} = \frac{x}{1.3 - x}\]
Since \((A \cap B' \cap C) \subseteq (A \cup C)\) and \(\dfrac{x}{1.3-x}\) is increasing on \([0.3, 0.5]\):
\((A \cap B' \cap C)\) is a subset of \((A \cup C)\), so \(\mathrm{P}\bigl((A \cap B' \cap C) \cap (A \cup C)\bigr) = \mathrm{P}(A \cap B' \cap C)\). Do NOT write \(\mathrm{P}(A \cap B' \cap C) \times \mathrm{P}(A \cup C)\) as they may not be independent.
(b)
\(A\) and \(C\) independent: \(\mathrm{P}(A \cap C) = \mathrm{P}(A)\cdot\mathrm{P}(C) = 0.8 \times 0.6 = 0.48\).
\(\mathrm{P}(A \cap C) = 0.1 + x = 0.48 \Rightarrow x = 0.38\).
Answer: (a)(i) \(\min x = 0.3\), \(\max x = 0.5\) (a)(ii) \(\min = 0.3\), \(\max = 0.625\) (b) \(x = 0.38\), \(\mathrm{P}(A \cap B' \cap C) = 0.38\)