Given that the events E and F are such that `P(E) = (1)/(2), P(E cup F) = (3)/(5)` and `P(F) = p`. Find the p, if E and F are independent events.

Answer

To solve the problem, we need to find the value of \( p \) given that events \( E \) and \( F \) are independent. We have the following information:

1. \( P(E) = \frac{1}{2} \)
2. \( P(E \cup F) = \frac{3}{5} \)
3. \( P(F) = p \)

Since \( E \) and \( F \) are independent events, we can use the formula for the probability of the union of two events:

\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]

For independent events, the probability of the intersection can be expressed as:

\[ P(E \cap F) = P(E) \cdot P(F) \]

Substituting the values we have:

\[ P(E \cap F) = P(E) \cdot P(F) = \left(\frac{1}{2}\right) \cdot p \]

Now, substituting this back into the union formula:

\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]

This gives us:

\[ \frac{3}{5} = \frac{1}{2} + p - \left(\frac{1}{2} \cdot p\right) \]

Now, let's simplify this equation step by step.

### Step 1: Substitute known values into the equation

Substituting the known values into the equation:

\[ \frac{3}{5} = \frac{1}{2} + p - \frac{1}{2}p \]

### Step 2: Combine like terms

Rearranging the equation gives:

\[ \frac{3}{5} = \frac{1}{2} + p - \frac{1}{2}p \]

This simplifies to:

\[ \frac{3}{5} = \frac{1}{2} + \frac{1}{2}p \]

### Step 3: Isolate \( p \)

To isolate \( p \), we first subtract \( \frac{1}{2} \) from both sides:

\[ \frac{3}{5} - \frac{1}{2} = \frac{1}{2}p \]

### Step 4: Find a common denominator

The common denominator for \( \frac{3}{5} \) and \( \frac{1}{2} \) is 10. Thus:

\[ \frac{3}{5} = \frac{6}{10} \quad \text{and} \quad \frac{1}{2} = \frac{5}{10} \]

So we have:

\[ \frac{6}{10} - \frac{5}{10} = \frac{1}{2}p \]

This simplifies to:

\[ \frac{1}{10} = \frac{1}{2}p \]

### Step 5: Solve for \( p \)

Now, multiply both sides by 2 to solve for \( p \):

\[ p = 2 \cdot \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \]

Thus, the value of \( p \) is:

\[ \boxed{\frac{1}{5}} \]