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Question
The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3, and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.
Solution
\[\text{ Suppose S represents the event of buying a shirt and T represents the event of buying a trouser } . \]
\[\text{ We have } , \]
\[P\left( S \right) = 0 . 2\]
\[P\left( T \right) = 0 . 3 \]
\[P\left( S/T \right) = 0 . 4\]
\[\text{ Now} , \]
\[P\left( S/T \right) = \frac{P\left( S \cap T \right)}{P\left( T \right)}\]
\[ \Rightarrow P\left( S \cap T \right) = P\left( S/T \right) \times P\left( T \right) = 0 . 4 \times 0 . 3 = 0 . 12\]
\[P\left( T/S \right) = \frac{P\left( S \cap T \right)}{P\left( S \right)} = \frac{0 . 12}{0 . 2} = 0 . 6\]
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