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Question
The probability of student A passing an examination is 2/9 and of student B passing is 5/9. Assuming the two events : 'A passes', 'B passes' as independent, find the probability of : (i) only A passing the examination (ii) only one of them passing the examination.
Solution
\[P\left( \text{ A passing examination } \right) = \frac{2}{9}\]
\[P\left( \text{ B passing examination } \right) = \frac{5}{9}\]
\[\left( i \right) P\left( \text{ only A passing examination} \right) = P\left( \text{ A passes} \right) P\left( B \text{ fails } \right)\]
\[ = \frac{2}{9}\left( 1 - \frac{5}{9} \right)\]
\[ = \frac{2}{9} \times \frac{4}{9}\]
\[ = \frac{8}{81}\]
\[\left( ii \right) P\left( \text{ only one of them passing examination } \right) = P\left( A\text{ passes and B fails }\right) + P\left( \text{ B passes and A fails } \right)\]
\[ = \frac{2}{9} \times \left( 1 - \frac{5}{9} \right) + \frac{5}{9} \times \left( 1 - \frac{2}{9} \right)\]
\[ = \frac{8}{81} + \frac{35}{81}\]
\[ = \frac{43}{81}\]
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