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Question
X is taking up subjects - Mathematics, Physics and Chemistry in the examination. His probabilities of getting grade A in these subjects are 0.2, 0.3 and 0.5 respectively. Find the probability that he gets
(i) Grade A in all subjects
(ii) Grade A in no subject
(iii) Grade A in two subjects.
Solution
\[P\left( \text{ A grade in Maths } \right) = P\left( A \right) = 0 . 2\]
\[P\left( \text{ A grade in Physics } \right) = P\left( B \right) = 0 . 3\]
\[P\left(\text{ A grade in Chemistry } \right) = P\left( C \right) = 0 . 5\]
\[\left( i \right) P\left( \text{ grade A in all subjects } \right) = P\left( A \right) \times P\left( B \right) \times P\left( C \right)\]
\[ = 0 . 2 \times 0 . 3 \times 0 . 5\]
\[ = 0 . 03\]
\[\left( ii \right) P\left( \text{ grade A in no subject } \right) = P\left( \bar{A} \right) \times P\left( \bar{B} \right) \times P\left( \bar{C} \right)\]
\[ = \left( 1 - 0 . 2 \right) \times \left( 1 - 0 . 3 \right) \times \left( 1 - 0 . 5 \right)\]
\[ = 0 . 8 \times 0 . 7 \times 0 . 5\]
\[ = 0 . 28\]
\[\left( iii \right) P\left( \text{ grade A in two subjects } \right) = P\left( \text{ not grade A in Maths } \right) + P\left( \text{ not grade A in Physics }\right) + P\left( \text{ not grade A in Chemistry } \right)\]
\[ = P\left( \overline{ A } \right) \times P\left( B \right) \times P\left( C \right) + P\left( A \right) \times P\left( \overline{ B } \right) \times P\left(\overline{ C } \right) + P\left( A \right) \times P\left( B \right) \times P\left( C \right)\]
\[ = \left( 1 - 0 . 2 \right) \times 0 . 3 \times 0 . 5 + 0 . 2 \times \left( 1 - 0 . 3 \right) \times 0 . 5 + 0 . 2 \times 0 . 3 \times \left( 1 - 0 . 5 \right)\]
\[ = 0 . 8 \times 0 . 3 \times 0 . 5 + 0 . 2 \times 0 . 7 \times 0 . 5 + 0 . 2 \times 0 . 3 \times 0 . 5\]
\[ = 0 . 12 + 0 . 07 + 0 . 03\]
\[ = 0 . 22\]
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