An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered 2,3,3,4…12 is picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is $\frac{P}{792}$ then find P.

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answer is 193.

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Detailed Solution

Let $E_{1} =$ the toss reslt in a head
$E_{2} =$ the toss result in a tail
A = noted number is 7 or 8
$∴ P (A) = P (E_{1}) . P (A / E_{1}) + P (E_{2}) . P (A / E_{2})$
$= \frac{1}{2} \times \frac{11}{36} + \frac{1}{2} \times \frac{2}{11} = \frac{193}{792}$
$∴ \frac{P}{792} = \frac{193}{792}$
$∴ P = 193$

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