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. `(15)/(72)`
B. `(13)/(36)`
C. `(19)/(72)`
D. `(19)/(36)`

Explanation

To solve the problem step by step, we will break down the events and calculate the required probabilities.

### Step 1: Understand the Events
We have two events based on the outcome of tossing an unbiased coin:
- If the coin shows **Heads (H)**, we roll a pair of unbiased dice.
- If the coin shows **Tails (T)**, we pick a card from a pack of 9 cards numbered 1 to 9.

### Step 2: Calculate the Probability of Coin Toss
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 Rolling Dice
Let \( E_1 \) be the event of getting a sum of 7 or 8 when rolling two dice.
- The total outcomes when rolling two dice = \( 6 \times 6 = 36 \).
- The favorable outcomes for a sum of 7:
- (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes
- The favorable outcomes for a sum of 8:
- (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes
- Total favorable outcomes for \( E_1 \) = 6 + 5 = 11.

Thus, the probability of getting a sum of 7 or 8 when rolling two dice is:
\[ P(E_1) = \frac{11}{36} \]

### Step 4: Calculate the Probability of Picking a Card
Let \( E_2 \) be the event of picking a card that shows either 7 or 8.
- The total outcomes when picking a card = 9 (cards numbered 1 to 9).
- The favorable outcomes for picking either 7 or 8 = 2 (7 and 8).

Thus, the probability of picking a card that shows either 7 or 8 is:
\[ P(E_2) = \frac{2}{9} \]

### Step 5: Calculate the Overall Probability
We need to find the probability that the noted number is either 7 or 8, which can occur in two mutually exclusive ways:
1. The coin shows Heads and we get a sum of 7 or 8 from the dice.
2. The coin shows Tails and we pick a card that shows either 7 or 8.

Using the law of total probability:
\[ P(\text{Noted number is 7 or 8}) = P(H) \cdot P(E_1) + P(T) \cdot P(E_2) \]
Substituting the values:
\[ P(\text{Noted number is 7 or 8}) = \left(\frac{1}{2} \cdot \frac{11}{36}\right) + \left(\frac{1}{2} \cdot \frac{2}{9}\right) \]

Calculating each term:
1. \( P(H) \cdot P(E_1) = \frac{1}{2} \cdot \frac{11}{36} = \frac{11}{72} \)
2. \( P(T) \cdot P(E_2) = \frac{1}{2} \cdot \frac{2}{9} = \frac{1}{9} = \frac{8}{72} \) (converting to a common denominator)

Now, adding these probabilities:
\[ P(\text{Noted number is 7 or 8}) = \frac{11}{72} + \frac{8}{72} = \frac{19}{72} \]

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