If the odds against the occurrence of an event be `4 : 7`, find the probability of the occurrence of the event.

Answer

To find the probability of the occurrence of an event when the odds against it are given as \(4 : 7\), we can follow these steps:

### Step 1: Understand the Odds
The odds against the occurrence of an event are given as \(4 : 7\). This means that for every 4 times the event does not occur, it occurs 7 times.

### Step 2: Calculate Total Outcomes
The total outcomes can be calculated by adding the two parts of the odds:
\[ \text{Total Outcomes} = \text{Odds against} + \text{Odds for} = 4 + 7 = 11 \]

### Step 3: Determine the Probability of the Event
The probability of the occurrence of the event can be calculated using the formula:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]
Here, the number of favorable outcomes (the event occurring) is represented by the odds for, which is 7. Thus, we have:
\[ P(E) = \frac{7}{11} \]

### Step 4: Conclusion
Therefore, the probability of the occurrence of the event is:
\[ \boxed{\frac{7}{11}} \]

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