If E be an event such that `P ( E ) = ( 3)/( 7)` , what is P( not E ) equal to ?

Answer

To find the probability of the event not occurring, denoted as P(not E), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information:**
We are given that the probability of event E is \( P(E) = \frac{3}{7} \).

2. **Use the Complement Rule:**
The probability of the event not occurring, \( P(not E) \), can be calculated using the formula:
\[ P(E) + P(not E) = 1 \]
This means that the sum of the probabilities of an event and its complement is always equal to 1.

3. **Substitute the Known Value:**
Substitute the value of \( P(E) \) into the equation:
\[ \frac{3}{7} + P(not E) = 1 \]

4. **Isolate \( P(not E) \):**
To find \( P(not E) \), subtract \( \frac{3}{7} \) from both sides:
\[ P(not E) = 1 - \frac{3}{7} \]

5. **Convert 1 to a Fraction:**
To perform the subtraction, convert 1 into a fraction with a denominator of 7:
\[ 1 = \frac{7}{7} \]
So now the equation looks like:
\[ P(not E) = \frac{7}{7} - \frac{3}{7} \]

6. **Perform the Subtraction:**
Now, subtract the fractions:
\[ P(not E) = \frac{7 - 3}{7} = \frac{4}{7} \]

7. **Conclusion:**
Therefore, the probability of the event not occurring is:
\[ P(not E) = \frac{4}{7} \]

### Final Answer:
\[ P(not E) = \frac{4}{7} \]

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