The odds against a certain event are 5: 2 and the odds in favour of another independent event are 6:5. The probability that at least one of the events will happen, is

Options

A. `(25)/(77)`
B. `(52)/(77)`
C. `(12)/(77)`
D. `(65)/(77)`

Explanation

To solve the problem, we need to find the probability that at least one of two independent events will happen. We are given the odds against one event and the odds in favor of another event.

### Step-by-Step Solution:

1. **Understanding Odds Against Event A**:
- The odds against event A are given as 5:2. This means for every 5 failures, there are 2 successes.
- The probability of event A happening (P(A)) can be calculated as:
\[ P(A) = \frac{\text{Number of successes}}{\text{Total outcomes}} = \frac{2}{5 + 2} = \frac{2}{7} \]

2. **Understanding Odds In Favor of Event B**:
- The odds in favor of event B are given as 6:5. This means for every 6 successes, there are 5 failures.
- The probability of event B happening (P(B)) can be calculated as:
\[ P(B) = \frac{\text{Number of successes}}{\text{Total outcomes}} = \frac{6}{6 + 5} = \frac{6}{11} \]

3. **Calculating the Probability of Neither Event Happening**:
- The probability of event A not happening (P(A')) is:
\[ P(A') = 1 - P(A) = 1 - \frac{2}{7} = \frac{5}{7} \]
- The probability of event B not happening (P(B')) is:
\[ P(B') = 1 - P(B) = 1 - \frac{6}{11} = \frac{5}{11} \]

4. **Finding the Probability of Neither Event Happening**:
- Since the events are independent, the probability that neither event A nor event B happens (P(A' ∩ B')) is:
\[ P(A' ∩ B') = P(A') \times P(B') = \frac{5}{7} \times \frac{5}{11} = \frac{25}{77} \]

5. **Calculating the Probability of At Least One Event Happening**:
- The probability that at least one of the events happens (P(A ∪ B)) is given by:
\[ P(A ∪ B) = 1 - P(A' ∩ B') = 1 - \frac{25}{77} = \frac{77 - 25}{77} = \frac{52}{77} \]

### Final Answer:
The probability that at least one of the events will happen is:
\[ \frac{52}{77} \]