If E and F are two events such that `P(E) =(1)/(4) ,P(F) = (1)/(2) and P(E nn F) = (1)/(8) , `then find `P( E' nn F')`

Answer

To solve the problem, we need to find \( P(E' \cap F') \) given the probabilities of events \( E \) and \( F \) and their intersection. Let's break it down step by step.

### Step 1: Write down the given probabilities.
We have:
- \( P(E) = \frac{1}{4} \)
- \( P(F) = \frac{1}{2} \)
- \( P(E \cap F) = \frac{1}{8} \)

### Step 2: Use the formula for the union of two events.
We need to find \( P(E \cup F) \). The formula for the probability of the union of two events is:
\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]

### Step 3: Substitute the values into the formula.
Substituting the known values:
\[ P(E \cup F) = \frac{1}{4} + \frac{1}{2} - \frac{1}{8} \]

### Step 4: Find a common denominator and calculate.
The least common multiple (LCM) of 4, 2, and 8 is 8. We can convert each term:
- \( P(E) = \frac{1}{4} = \frac{2}{8} \)
- \( P(F) = \frac{1}{2} = \frac{4}{8} \)
- \( P(E \cap F) = \frac{1}{8} \)

Now substituting these values:
\[ P(E \cup F) = \frac{2}{8} + \frac{4}{8} - \frac{1}{8} = \frac{2 + 4 - 1}{8} = \frac{5}{8} \]

### Step 5: Use De Morgan's Law to find \( P(E' \cap F') \).
According to De Morgan's Law:
\[ P(E' \cap F') = 1 - P(E \cup F) \]

### Step 6: Substitute the value of \( P(E \cup F) \).
Now substituting the value we found:
\[ P(E' \cap F') = 1 - \frac{5}{8} \]

### Step 7: Calculate the final result.
\[ P(E' \cap F') = \frac{8}{8} - \frac{5}{8} = \frac{3}{8} \]

### Conclusion
Thus, the final answer is:
\[ P(E' \cap F') = \frac{3}{8} \]

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