If the difference between the probability of happening & non happening of an event E is `(3)/(7)` Find the probability of happening of the event E.

Options

A. `(5)/(7)`
B. `(2)/(7)`
C. `(3)/(7)`
D. `(4)/(7)`

Answer

To find the probability of happening of the event E, we can follow these steps:

### Step 1: Define the probabilities
Let \( P(E) \) be the probability of the event E happening. The probability of the event E not happening (the complement of E) is given by:
\[ P(E') = 1 - P(E) \]

### Step 2: Set up the equation
According to the problem, the difference between the probability of happening and non-happening of event E is given as \( \frac{3}{7} \). Therefore, we can write the equation as:
\[ P(E) - P(E') = \frac{3}{7} \]

### Step 3: Substitute the complement
Substituting \( P(E') \) into the equation, we have:
\[ P(E) - (1 - P(E)) = \frac{3}{7} \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ P(E) - 1 + P(E) = \frac{3}{7} \]
\[ 2P(E) - 1 = \frac{3}{7} \]

### Step 5: Solve for \( P(E) \)
Next, we isolate \( P(E) \):
\[ 2P(E) = \frac{3}{7} + 1 \]
To add \( 1 \) (which is \( \frac{7}{7} \)):
\[ 2P(E) = \frac{3}{7} + \frac{7}{7} = \frac{10}{7} \]
Now, divide both sides by 2:
\[ P(E) = \frac{10}{7} \times \frac{1}{2} = \frac{10}{14} = \frac{5}{7} \]

### Conclusion
The probability of happening of the event E is:
\[ P(E) = \frac{5}{7} \]

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