1. Key Concepts and Formulas

• Law of Total Probability: If $E_1, E_2, \ldots, E_n$ are a set of mutually exclusive and exhaustive events (meaning one of them must occur, and no two can occur simultaneously), then the probability of any event $A$ can be calculated as the sum of the probabilities of $A$ occurring under each of the events $E_i$: $P(A) = P(A|E_1)P(E_1) + P(A|E_2)P(E_2) + \ldots + P(A|E_n)P(E_n)$ Here, $P(A|E_i)$ is the conditional probability of event $A$ occurring given that event $E_i$ has occurred, and $P(E_i)$ is the probability of event $E_i$ occurring.
• Calculating Probability for Equally Likely Outcomes: For an event in a sample space with equally likely outcomes, the probability is given by: $P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

2. Step-by-Step Solution

Step 1: Define Events and Their Initial Probabilities First, let's clearly define the events involved in this problem to avoid any confusion:

• Let $H$ be the event that the coin toss results in a Head.
• Let $T$ be the event that the coin toss results in a Tail.
• Let $A$ be the event that the noted number (which is either the sum from the dice roll or the number on the picked card) is either 7 or 8.

Since the coin is unbiased, the probabilities of getting a Head or a Tail are straightforward:

• $P(H) = \frac{1}{2}$
• $P(T) = \frac{1}{2}$

Step 2: Calculate Conditional Probability for the "Heads" Scenario ($P(A|H)$) If the coin toss results in a Head, a pair of unbiased dice is rolled. We need to determine the probability that the sum of the numbers obtained on these dice is either 7 or 8.

• Total possible outcomes when rolling two dice: Each die has 6 faces, so when two dice are rolled, the total number of distinct and equally likely outcomes is $6 \times 6 = 36$.
• Favorable outcomes for a sum of 7: The pairs of numbers $(die_1, die_2)$ that add up to 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 such outcomes.
• Favorable outcomes for a sum of 8: The pairs of numbers $(die_1, die_2)$ that add up to 8 are: (2,6), (3,5), (4,4), (5,3), (6,2). There are 5 such outcomes.
• Total favorable outcomes for a sum of 7 or 8: Since getting a sum of 7 and getting a sum of 8 are mutually exclusive events (they cannot happen at the same time), we simply add the number of outcomes for each: $6 + 5 = 11$.
• Conditional Probability $P(A|H)$: $P(A|H) = \frac{\text{Number of favorable outcomes for sum 7 or 8}}{\text{Total possible outcomes for two dice}} = \frac{11}{36}$

Step 3: Calculate Conditional Probability for the "Tails" Scenario ($P(A|T)$) If the coin toss results in a Tail, a card is picked from a well-shuffled pack of nine cards numbered 1, 2, 3, ……, 9. We need to find the probability that the number on the picked card is either 7 or 8.

• Total possible outcomes when picking a card: There are 9 cards, numbered 1 through 9. Thus, the total number of equally likely outcomes is 9.
• Favorable outcomes for the card number being 7 or 8: The numbers on the cards that are either 7 or 8 are {7, 8}. There are 2 such outcomes.
• Conditional Probability $P(A|T)$: $P(A|T) = \frac{\text{Number of favorable outcomes for card 7 or 8}}{\text{Total possible outcomes for card pick}} = \frac{2}{9}$

Step 4: Combine Probabilities to Find the Overall Probability The problem asks for the overall probability that the noted number is either 7 or 8. To find this, we combine the probabilities from the two scenarios. $P(A) = P(A|H) + P(A|T)$ Now, we substitute the conditional probabilities calculated in the previous steps: $P(A) = \frac{11}{36} + \frac{2}{9}$ To add these fractions, we need a common denominator. The least common multiple of 36 and 9 is 36. We convert $\frac{2}{9}$ to an equivalent fraction with a denominator of 36: $\frac{2}{9} = \frac{2 \times 4}{9 \times 4} = \frac{8}{36}$ Now, substitute this back into the equation for $P(A)$: $P(A) = \frac{11}{36} + \frac{8}{36}$ $P(A) = \frac{11 + 8}{36}$ $P(A) = \frac{19}{36}$

3. Common Mistakes & Tips

• Clearly Define Events: Always start by defining your events ($H, T, A$ in this case) to ensure clarity throughout your solution.
• Understand Conditional Probability: Recognize when you need to calculate $P(A|E)$ – the probability of an event given that another event has already occurred.
• Systematic Listing for Dice Problems: For problems involving dice, systematically listing all favorable outcomes (e.g., for sums of 7 and 8) is crucial to avoid missing any possibilities. Remember that (1,6) is a distinct outcome from (6,1).
• Mutually Exclusive Events: When calculating the probability of "A or B" where A and B cannot happen simultaneously (like a sum of 7 and a sum of 8 on dice), you add their individual probabilities or counts of outcomes.

4. Summary

This problem required us to calculate the probability of a specific outcome (a noted number being 7 or 8) which could result from one of two different random experiments, chosen by an initial coin toss. We systematically determined the probability of the desired outcome in each of these conditional scenarios. Finally, by combining these probabilities, we found the overall probability that the noted number is either 7 or 8. The final calculated probability is $\frac{19}{36}$.

5. Final Answer

The final answer is $\boxed{\frac{19}{36}}$, which corresponds to option (A).