If P (E) = 0.40, P (F) = 0.35, P (E`uu`F) = 0.55, then P(F/E) is

Options

A. `0.5`
B. `0.25`
C. `0.6`
D. `0.7`

Answer

To solve the problem, we need to find \( P(F|E) \) using the given probabilities. The formula for conditional probability is:

\[ P(F|E) = \frac{P(E \cap F)}{P(E)} \]

### Step 1: Identify the given probabilities
We have:
- \( P(E) = 0.40 \)
- \( P(F) = 0.35 \)
- \( P(E \cup F) = 0.55 \)

### Step 2: Use the formula for the union of two events
We know that:

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

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula gives us:

\[ 0.55 = 0.40 + 0.35 - P(E \cap F) \]

### Step 4: Solve for \( P(E \cap F) \)
Rearranging the equation to find \( P(E \cap F) \):

\[ P(E \cap F) = 0.40 + 0.35 - 0.55 \]
\[ P(E \cap F) = 0.75 - 0.55 \]
\[ P(E \cap F) = 0.20 \]

### Step 5: Calculate \( P(F|E) \)
Now we can use the value of \( P(E \cap F) \) to find \( P(F|E) \):

\[ P(F|E) = \frac{P(E \cap F)}{P(E)} = \frac{0.20}{0.40} \]

### Step 6: Simplify the fraction
\[ P(F|E) = 0.5 \]

### Final Answer
Thus, \( P(F|E) = 0.5 \).

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