An experiment has 10 equally likely outcomes. Let $A$ and $B$ be two non-empty events of the experiment. If…

An experiment has 10 equally likely outcomes. Let $A$ and $B$ be two non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is
  1. 2,4 or 8
  2. 3,6 or 9
  3. 4 or 8
  4. 5 or 10

Solution

$P(A)=\frac{2}{5}$ For independent events, $ \begin{aligned} & P(A \cap B)=P(A) P(B) \Rightarrow P(A \cap B) \leq \frac{2}{5} \\ & \Rightarrow P(A \cap B)=\frac{1}{10}, \frac{2}{10}, \frac{3}{10}, \frac{4}{10} \\ & \text { (i) } P(A \cap B)=\frac{1}{10} \\ & \Rightarrow P(A) \cdot P(B)=\frac{1}{10} \\ & \Rightarrow \quad P(B)=\frac{1}{10} \times \frac{5}{2}=\frac{1}{4}, \text { not possible. } \\ & \text { (ii) } P(A \cap B)=\frac{2}{10} \Rightarrow \frac{2}{5} \times P(B)=\frac{2}{10} \\ & \Rightarrow \quad P(B)=\frac{5}{10}, \text { outcomes of } B=5 \end{aligned} $ $ \text { (iii) } \begin{aligned} & P(A \cap B)=\frac{3}{10} \Rightarrow P(A) \cdot P(B)=\frac{3}{10} \\ & \Rightarrow \quad \frac{2}{5} \times P(B)=\frac{3}{10} \\ & P(B)=\frac{3}{4}, \text { not possible } \end{aligned} $ $ \text { (iv) } \begin{aligned} & P(A \cap B)=\frac{4}{10} \\ \Rightarrow & P(A) \cdot P(B)=\frac{4}{10} \\ \Rightarrow & P(B)=1, \text { outcomes of } B=10 . \end{aligned} $

Asked in: JEE Advanced 2008 (Paper 2)

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