One ticket is selected at random from 50 tickets numbered 00, 01, 02, ... , 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals (1) 1/14 (2) 1/7 (3) 5/14 (4) 1/50

Options

A. `1/14`
B. `1/7`
C. `5/14`
D. `1/50`

Explanation

To solve the problem, we need to find the probability that the sum of the digits on a randomly selected ticket is 8, given that the product of those digits is 0. Let's break it down step by step.

### Step 1: Understand the Total Tickets
The tickets are numbered from 00 to 49, which gives us a total of 50 tickets.

### Step 2: Identify the Condition
We are given that the product of the digits is 0. This means at least one of the digits must be 0. The two-digit tickets can be represented as XY, where X is the tens digit and Y is the units digit.

### Step 3: List the Tickets with Product of Digits Equal to 0
The tickets that have at least one digit as 0 are:
- 00, 01, 02, 03, 04, 05, 06, 07, 08, 09 (10 tickets)
- 10, 20, 30, 40 (4 tickets)
- 11, 12, 13, 14, 15, 16, 17, 18, 19 (not included as they do not have a 0)

Thus, the total tickets where the product of the digits is 0 is:
- Total = 10 (from 00 to 09) + 4 (from 10, 20, 30, 40) = 14 tickets.

### Step 4: Identify Tickets with Sum of Digits Equal to 8
Next, we need to find which of these tickets also have a sum of digits equal to 8:
- 08 (0 + 8 = 8)
- 17 (1 + 7 = 8)
- 26 (2 + 6 = 8)
- 35 (3 + 5 = 8)
- 44 (4 + 4 = 8)

However, only the tickets that also have a product of 0 are valid. From the above, the only ticket that meets both conditions (product of digits = 0 and sum of digits = 8) is:
- 08

### Step 5: Calculate the Required Probability
Now we can calculate the probability using the formula:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]

Where:
- \( P(A \cap B) \) is the probability of both events happening (sum = 8 and product = 0). There is 1 favorable outcome (08).
- \( P(B) \) is the probability of the product of digits being 0. There are 14 outcomes.

Thus, the probability is:
\[ P(A | B) = \frac{1}{14} \]

### Final Answer
The probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is 0, is \( \frac{1}{14} \).