One ticket is selected at random from 100 tickets numbered `00,01,02, …, 99.` Suppose A and B are the sum and product of the digit found on the ticket, respectively. Then `P((A=7)//(B=0))` is given by

Options

A. `2//13`
B. `2//19`
C. `1//50`
D. None of these

Explanation

To solve the problem, we need to find the probability \( P(A = 7 | B = 0) \), where \( A \) is the sum of the digits on a randomly selected ticket and \( B \) is the product of the digits on that ticket.

### Step-by-step Solution:

1. **Identify the Sample Space**:
The tickets are numbered from 00 to 99, giving us a total of 100 tickets.

2. **Determine the Condition for \( B = 0 \)**:
The product \( B \) will be 0 if at least one of the digits is 0. The possible tickets satisfying \( B = 0 \) are:
- Tickets with the first digit as 0: 00, 01, 02, ..., 09 (10 tickets)
- Tickets with the second digit as 0: 10, 20, 30, ..., 90 (9 tickets)
- The ticket 00 is counted in both cases, so we need to subtract it once.

Thus, the total number of tickets where \( B = 0 \) is:
\[ 10 + 9 - 1 = 18 \]
Therefore, \( P(B = 0) = \frac{18}{100} = \frac{9}{50} \).

3. **Determine the Condition for \( A = 7 \)**:
The sum \( A \) will equal 7 for the following pairs of digits:
- (0, 7) → 07
- (1, 6) → 16
- (2, 5) → 25
- (3, 4) → 34
- (4, 3) → 43
- (5, 2) → 52
- (6, 1) → 61
- (7, 0) → 70

Thus, the total number of tickets where \( A = 7 \) is 8, so \( P(A = 7) = \frac{8}{100} = \frac{2}{25} \).

4. **Find the Intersection \( P(A = 7 \cap B = 0) \)**:
We need to find the tickets that satisfy both \( A = 7 \) and \( B = 0 \). The valid tickets from our previous findings are:
- 07 (A = 7, B = 0)
- 70 (A = 7, B = 0)

Thus, there are 2 tickets that satisfy both conditions, so:
\[ P(A = 7 \cap B = 0) = \frac{2}{100} = \frac{1}{50} \]

5. **Apply the Conditional Probability Formula**:
Using the formula for conditional probability:
\[ P(A = 7 | B = 0) = \frac{P(A = 7 \cap B = 0)}{P(B = 0)} \]
Substituting the values we found:
\[ P(A = 7 | B = 0) = \frac{\frac{1}{50}}{\frac{9}{50}} = \frac{1}{9} \]

### Final Answer:
Thus, the probability \( P(A = 7 | B = 0) \) is \( \frac{1}{9} \).