Question:

Step 1: Determine the probabilities of the coin toss. The coin is unbiased, so the probability of getting a Head, $P(H)$, and the probability of getting a Tail, $P(T)$, are: $$P(H) = \frac{1}{2}$$ $$P(T) = \frac{1}{2}$$ Step 2: Calculate the probability of noting a 7 or 8 if the result is a Head. If the coin shows a Head, a pair of unbiased dice is rolled. The total number of possible outcomes when rolling two dice is $6 \times 6 = 36$. The combinations that sum to 7 are: $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$, which are 6 ways. The combinations that sum to 8 are: $(2,6), (3,5), (4,4), (5,3), (6,2)$, which are 5 ways. The total number of ways to get a sum of 7 or 8 is $6 + 5 = 11$ ways. The conditional probability of noting a 7 or 8 given a Head, $P(7 \text{ or } 8 | H)$, is: $$P(7 \text{ or } 8 | H) = \frac{11}{36}$$ Step 3: Calculate the probability of noting a 7 or 8 if the result is a Tail. If the coin shows a Tail, a card is picked from a well-shuffled pack of eleven cards numbered 2, 3, 4, ..., 12. The total number of possible outcomes is 11. The cards showing 7 or 8 are {7, 8}, which are 2 cards. The conditional probability of noting a 7 or 8 given a Tail, $P(7 \text{ or } 8 | T)$, is: $$P(7 \text{ or } 8 | T) = \frac{2}{11}$$ Step 4: Calculate the total probability that the noted number is either 7 or 8. Using the law of total probability, the probability that the noted number is either 7 or 8, $P(7 \text{ or } 8)$, is: $$P(7 \text{ or } 8) = P(H) \times P(7 \text{ or } 8 | H) + P(T) \times P(7 \text{ or } 8 | T)$$ Substituting the values from the previous steps: $$P(7 \text{ or } 8) = \left(\frac{1}{2}\right) \left(\frac{11}{36}\right) + \left(\frac{1}{2}\right) \left(\frac{2}{11}\right)$$ $$P(7 \text{ or } 8) = \frac{11}{72} + \frac{2}{22}$$ $$P(7 \text{ or } 8) = \frac{11}{72} + \frac{1}{11}$$ To sum these fractions, find a common denominator, which is $72 \times 11 = 792$: $$P(7 \text{ or } 8) = \frac{11 \times 11}{72 \times 11} + \frac{1 \times 72}{11 \times 72}$$ $$P(7 \text{ or } 8) = \frac{121}{792} + \frac{72}{792}$$ $$P(7 \text{ or } 8) = \frac{121 + 72}{792}$$ $$P(7 \text{ or } 8) = \frac{193}{792}$$