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Question
An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the sum of the numbers obtained is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered 2, 3, 4, ..., 12 is picked and the number on the card is noted. What is the probability that the noted number is either 7 or 8?
Solution
Let E1, E2 and A be the events as defined below:
E1 = The coin shows a head
E2 = The coin shows a head
A = The noted number is 7 or 8
\[\therefore P\left( E_1 \right) = \frac{1}{2} \]
\[ P\left( E_2 \right) = \frac{1}{2}\]
\[\text{ Now } , \]
\[P\left( A/ E_1 \right) = \frac{11}{36}\]
\[P\left( A/ E_2 \right) = \frac{2}{11}\]
\[\text{ Using the law of total probability, we get } \]
\[\text{ Required probability } = P\left( A \right) = P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right)\]
\[ = \frac{1}{2} \times \frac{11}{36} + \frac{1}{2} \times \frac{2}{11}\]
\[ = \frac{11}{72} + \frac{1}{11}\]
\[ = \frac{121 + 72}{792} = \frac{193}{792}\]
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