A lottery consists 3 prize tickets and 6 blank tickets. A choose three tickets from this lottery . Another lottery there are 1 prize ticket and 2 blank tickets. B choose one ticket from this lottery. Find the ratio in favour of A and B.

Options

A. `7:16`
B. `16:7`
C. `16:9`
D. `1:28`

Explanation

To solve the problem, we need to find the probabilities of A and B winning in their respective lotteries and then determine the ratio of these probabilities.

### Step-by-Step Solution:

1. **Understanding Lottery A:**
- Lottery A consists of 3 prize tickets and 6 blank tickets, making a total of 9 tickets.
- A chooses 3 tickets from this lottery.
- We need to find the probability that A picks only prize tickets.

2. **Calculating the Probability for A:**
- The total number of ways to choose 3 tickets from 9 is given by the combination formula:
\[ \text{Total ways} = \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]
- The number of favorable outcomes (choosing all 3 prize tickets) is:
\[ \text{Favorable ways} = \binom{3}{3} = 1 \]
- Therefore, the probability \( P(A) \) that A picks only prize tickets is:
\[ P(A) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{84} \]

3. **Understanding Lottery B:**
- Lottery B consists of 1 prize ticket and 2 blank tickets, making a total of 3 tickets.
- B chooses 1 ticket from this lottery.
- We need to find the probability that B picks the prize ticket.

4. **Calculating the Probability for B:**
- The number of favorable outcomes for B (choosing the prize ticket) is 1.
- The total number of tickets is 3.
- Therefore, the probability \( P(B) \) that B picks the prize ticket is:
\[ P(B) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{3} \]

5. **Finding the Ratio of Probabilities:**
- We need to find the ratio \( P(A) : P(B) \):
\[ \text{Ratio} = \frac{P(A)}{P(B)} = \frac{\frac{1}{84}}{\frac{1}{3}} = \frac{1}{84} \times \frac{3}{1} = \frac{3}{84} = \frac{1}{28} \]
- Thus, the ratio in favor of A and B is:
\[ P(A) : P(B) = 1 : 28 \]

### Final Answer:
The ratio in favor of A and B is \( 1 : 28 \).