Three candidates solve a question. Odds in favour of the correct answer are 5:2, 4:3 and 3:4 respectively for the three candidates. What is the probability that at least two of them solve the question correctly?

Options

A. A. `(209)/(243)`
B. B. `(134)/(343)`
C. C. `(149)/(343)`
D. D. `(60)/(343)`

Answer

To solve the problem, we need to find the probability that at least two out of three candidates (A, B, and C) solve a question correctly, given their odds in favor of answering correctly.

### Step-by-Step Solution:

1. **Understanding Odds**:
- The odds in favor of candidate A answering correctly are 5:2. This means for every 5 successes, there are 2 failures.
- The odds in favor of candidate B answering correctly are 4:3.
- The odds in favor of candidate C answering correctly are 3:4.

2. **Calculating Probabilities**:
- For candidate A:
\[ P(A) = \frac{5}{5 + 2} = \frac{5}{7} \]
- For candidate B:
\[ P(B) = \frac{4}{4 + 3} = \frac{4}{7} \]
- For candidate C:
\[ P(C) = \frac{3}{3 + 4} = \frac{3}{7} \]

3. **Finding the Probability of At Least Two Candidates Answering Correctly**:
- The event "at least two candidates answer correctly" can be broken down into three cases:
1. Exactly two candidates answer correctly.
2. All three candidates answer correctly.

4. **Calculating the Probability for Each Case**:
- **Case 1: Exactly A and B answer correctly, C does not**:
\[ P(A \cap B \cap C') = P(A) \cdot P(B) \cdot (1 - P(C)) = \frac{5}{7} \cdot \frac{4}{7} \cdot \left(1 - \frac{3}{7}\right) = \frac{5}{7} \cdot \frac{4}{7} \cdot \frac{4}{7} = \frac{80}{343} \]

- **Case 2: Exactly A and C answer correctly, B does not**:
\[ P(A \cap B' \cap C) = P(A) \cdot (1 - P(B)) \cdot P(C) = \frac{5}{7} \cdot \left(1 - \frac{4}{7}\right) \cdot \frac{3}{7} = \frac{5}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} = \frac{45}{343} \]

- **Case 3: Exactly B and C answer correctly, A does not**:
\[ P(A' \cap B \cap C) = (1 - P(A)) \cdot P(B) \cdot P(C) = \left(1 - \frac{5}{7}\right) \cdot \frac{4}{7} \cdot \frac{3}{7} = \frac{2}{7} \cdot \frac{4}{7} \cdot \frac{3}{7} = \frac{24}{343} \]

- **Case 4: All three candidates answer correctly**:
\[ P(A \cap B \cap C) = P(A) \cdot P(B) \cdot P(C) = \frac{5}{7} \cdot \frac{4}{7} \cdot \frac{3}{7} = \frac{60}{343} \]

5. **Adding Probabilities**:
- Now, we add the probabilities of all cases where at least two candidates answer correctly:
\[ P(\text{at least 2 correct}) = P(A \cap B \cap C') + P(A \cap B' \cap C) + P(A' \cap B \cap C) + P(A \cap B \cap C) \]
\[ = \frac{80}{343} + \frac{45}{343} + \frac{24}{343} + \frac{60}{343} = \frac{209}{343} \]

### Final Probability:
Thus, the probability that at least two candidates solve the question correctly is:
\[ \boxed{\frac{209}{343}} \]