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Question
If S is the sample space and P (A) = \[\frac{1}{3}\]P (B) and S = A ∪ B, where A and B are two mutually exclusive events, then P (A) =
Options
1/4
1/2
3/4
3/8
Solution
\[\frac{1}{4}\]
\[P\left( A \right) = \frac{1}{3}P\left( B \right)\]
\[ \Rightarrow P\left( B \right) = 3P\left( A \right) . . . \left( 1 \right)\]
\[\text{ A and B are mutually exclusive events } .\]
\[\Rightarrow P\left( A \cap B \right) = 0\]
\[\text{ Now } ,\]
\[P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) = P\left( S \right)\]
\[ \Rightarrow P\left( A \right) + P\left( B \right) = 1\]
\[ \Rightarrow P\left( A \right) + 3P\left( A \right) = 1 \left[ \text{ From } \left( 1 \right) \right]\]
\[ \Rightarrow 4P\left( A \right) = 1\]
\[ \Rightarrow P\left( A \right) = \frac{1}{4}\]
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