E and F are two events associated with a random rxperiment for which P(F)=0.35,P(E or F) =0.85,P(E and F)=0.15 Find P€.

Answer

To solve the problem, we will use the formula for the probability of the union of two events. The formula states:

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

Where:
- \( P(E \cup F) \) is the probability of either event E or event F occurring.
- \( P(E) \) is the probability of event E occurring.
- \( P(F) \) is the probability of event F occurring.
- \( P(E \cap F) \) is the probability of both events E and F occurring.

Given:
- \( P(F) = 0.35 \)
- \( P(E \cup F) = 0.85 \)
- \( P(E \cap F) = 0.15 \)

We need to find \( P(E) \).

### Step 1: Write down the formula for the union of two events.

Using the formula:

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

### Step 2: Substitute the known values into the formula.

Substituting the values we have:

\[ 0.85 = P(E) + 0.35 - 0.15 \]

### Step 3: Simplify the equation.

Now, simplify the right side:

\[ 0.85 = P(E) + 0.20 \]

### Step 4: Isolate \( P(E) \).

To find \( P(E) \), we need to isolate it:

\[ P(E) = 0.85 - 0.20 \]

### Step 5: Calculate \( P(E) \).

Now, perform the subtraction:

\[ P(E) = 0.65 \]

### Final Answer:

Thus, the probability of event E is:

\[ P(E) = 0.65 \]

---