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 shufled pack of eleven cards numbered $2,3,4,...12$ is picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$, is

Using the law of total probability, the chance of getting a 7 or 8 is the sum of the probabilities from the dice and the cards. The probability of getting a head is 1/2, and rolling a sum of 7 or 8 on two dice yields 11 favorable outcomes out of 36, giving 11/72. The probability of getting a tail is 1/2, and picking a 7 or 8 from the 11 cards gives 2/11, so adding these branches gives (1/2) x (11/72) + (1/2) x (2/11) = 11/144 + 1/11. Finding a common denominator of 1584 gives 121/1584 + 144/1584 = 265/1584, which simplifies to 193/792.

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