An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the coin results in tail, then a card from a well-shuffled pack of nine cards numbered 1,2,3,…….9 is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is

Options

A. `13/16`
B. `16/72`
C. `19/72`
D. `19/36`

Explanation

To solve the problem, we need to calculate the probability that the noted number is either 7 or 8 based on the outcome of tossing an unbiased coin. Here's the step-by-step solution:

### Step 1: Understand the Events
1. **Event H**: The coin shows heads.
2. **Event T**: The coin shows tails.
3. **Event E1**: The sum of the numbers on the two dice is either 7 or 8.
4. **Event E2**: The noted number from the card is either 7 or 8.

### Step 2: Calculate the Probability of Each Coin Outcome
The probability of getting heads (H) or tails (T) when tossing an unbiased coin is:
- \( P(H) = \frac{1}{2} \)
- \( P(T) = \frac{1}{2} \)

### Step 3: Calculate the Probability of Event E1 (Sum of Dice)
When two unbiased dice are rolled, the total number of outcomes is \( 6 \times 6 = 36 \).

**Outcomes for E1 (sum is 7 or 8)**:
- **For sum = 7**: The pairs are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → Total = 6 outcomes.
- **For sum = 8**: The pairs are (2,6), (3,5), (4,4), (5,3), (6,2) → Total = 5 outcomes.

Total outcomes for E1:
- \( \text{Total for E1} = 6 + 5 = 11 \)

Thus, the probability of E1 is:
\[ P(E1) = \frac{11}{36} \]

### Step 4: Calculate the Probability of Event E2 (Card Draw)
There are 9 cards numbered 1 to 9. The favorable outcomes for E2 (noted number is 7 or 8) are:
- Cards: 7, 8 → Total = 2 outcomes.

Thus, the probability of E2 is:
\[ P(E2) = \frac{2}{9} \]

### Step 5: Calculate the Total Probability
Using the law of total probability:
\[ P(E) = P(H) \cdot P(E1) + P(T) \cdot P(E2) \]
Substituting the values:
\[ P(E) = \left(\frac{1}{2} \cdot \frac{11}{36}\right) + \left(\frac{1}{2} \cdot \frac{2}{9}\right) \]

### Step 6: Simplifying the Expression
Calculating each term:
1. First term:
\[ \frac{1}{2} \cdot \frac{11}{36} = \frac{11}{72} \]
2. Second term:
\[ \frac{1}{2} \cdot \frac{2}{9} = \frac{1}{9} = \frac{8}{72} \]

Adding these two results:
\[ P(E) = \frac{11}{72} + \frac{8}{72} = \frac{19}{72} \]

### Final Answer
The probability that the noted number is either 7 or 8 is:
\[ \boxed{\frac{19}{72}} \]