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Question
Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is 1/4 and that to Tarun's rejection is 2/3. Find the probability that at least one of them will be selected.
Solution
\[P\left( \text{ Arun gets selected } \right) = P\left( A \right) = \frac{1}{4}\]
\[P\left( \text{ Tarun gets rejected } \right) = P\left( \bar{B} \right) = \frac{2}{3}\]
\[ \Rightarrow P\left( \text{ Tarun gets selected } \right) = 1 - \frac{2}{3} = \frac{1}{3}\]
\[P\left(\text{ atleast one of them is selected } \right) = P\left( A \cup B \right)\]
\[ \Rightarrow P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right)\]
\[ = P\left( A \right) + P\left( B \right) - P\left( A \right) \times P\left( B \right)\]
\[ = \frac{1}{4} + \frac{1}{3} - \frac{1}{4} \times \frac{1}{3}\]
\[ = \frac{3 + 4 - 1}{12}\]
\[ = \frac{1}{2}\]
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