One ticket is selected at ransom form 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, is

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 the digits is zero. Let's break down the solution step by step.

### Step 1: Identify the Sample Space
The tickets are numbered from 00 to 49. This gives us a total of 50 tickets.

**Hint:** The sample space consists of all possible outcomes, which in this case are the 50 tickets numbered from 00 to 49.

### Step 2: Determine the Condition (Product of Digits is Zero)
The product of the digits of a ticket will be zero if at least one of the digits is zero. The tickets that satisfy this condition 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 (9 tickets)
- 21, 22, 23, 24, 25, 26, 27, 28, 29 (9 tickets)
- 31, 32, 33, 34, 35, 36, 37, 38, 39 (9 tickets)
- 41, 42, 43, 44, 45, 46, 47, 48, 49 (9 tickets)

Counting these, we find:
- 10 (from 00 to 09)
- 4 (from 10, 20, 30, 40)
- 9 (from 11 to 19)
- 9 (from 21 to 29)
- 9 (from 31 to 39)
- 9 (from 41 to 49)

Total tickets with product zero = 10 + 4 + 9 + 9 + 9 + 9 = 50

**Hint:** Count all tickets where at least one digit is zero to find the total tickets satisfying the condition.

### Step 3: Find the Tickets Where the Sum of Digits is 8
Next, we need to find the tickets where the sum of the digits equals 8. The possible pairs of digits (a, b) such that a + b = 8 and at least one digit is zero are:
- 08 (0 + 8)
- 17 (1 + 7)
- 26 (2 + 6)
- 35 (3 + 5)
- 44 (4 + 4)

However, we need to ensure that at least one digit is zero. The only ticket that satisfies both conditions (sum = 8 and product = 0) is:
- 08 (0 + 8)

Thus, there is only 1 ticket that satisfies both conditions.

**Hint:** List all pairs of digits that sum to 8 and check which ones have at least one digit as zero.

### Step 4: Calculate the Probability
Now, we can calculate the probability using the formula:
\[ P(E2 | E1) = \frac{P(E2 \cap E1)}{P(E1)} \]

Where:
- \( E1 \) is the event that the product of digits is zero.
- \( E2 \) is the event that the sum of the digits is 8.

From our previous steps:
- \( P(E1) = \frac{14}{50} \) (14 tickets where the product is zero)
- \( P(E2 \cap E1) = \frac{1}{50} \) (1 ticket where both conditions are satisfied)

Thus, the probability is:
\[ P(E2 | E1) = \frac{1/50}{14/50} = \frac{1}{14} \]

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

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