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Question
The probability that A hits a target is 1/3 and the probability that B hits it, is 2/5, What is the probability that the target will be hit, if each one of A and B shoots at the target?
Solution
\[P\left( A \right) = P\left( \text{ A hits target } \right) = \frac{1}{3}\]
\[P\left( B \right) = P\left( B \text{ hits target } \right) = \frac{2}{5}\]
\[\text{ Now } , \]
\[P\left( A \cup B \right) = P\left( \text{ target will be hit by either A or B } \right)\]
\[ \Rightarrow P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right) \]
\[ \Rightarrow P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \right)P\left( B \right) \left[ \text{ A and B are independent } \right]\]
\[ \Rightarrow P\left( A \cup B \right) = \frac{1}{3} + \frac{2}{5} - \frac{1}{3} \times \frac{2}{5}\]
\[ \Rightarrow P\left( A \cup B \right) = \frac{5 + 6 - 2}{15}\]
\[ \Rightarrow P\left( A \cup B \right) = \frac{9}{15}\]
\[ \Rightarrow P\left( A \cup B \right) = \frac{3}{5}\]
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