Suppose `E_1` and `E_2` are two events of a sample space such that `P(E_1)=1/2, P(E_2|E_1)=1/2, P(E_1|E_2)=1/4` then `P(E'_2)`= _____

Answer

To solve the problem, we need to find \( P(E'_2) \), which is the probability of the complement of event \( E_2 \). We are given the following probabilities:

- \( P(E_1) = \frac{1}{2} \)
- \( P(E_2 | E_1) = \frac{1}{2} \)
- \( P(E_1 | E_2) = \frac{1}{4} \)

Let's denote \( P(E_2) \) as \( p \). We can use the definitions of conditional probability to find \( P(E_2) \).

### Step 1: Find \( P(E_2 \cap E_1) \)

Using the definition of conditional probability:

\[ P(E_2 | E_1) = \frac{P(E_2 \cap E_1)}{P(E_1)} \]

Substituting the known values:

\[ \frac{1}{2} = \frac{P(E_2 \cap E_1)}{\frac{1}{2}} \]

Multiplying both sides by \( \frac{1}{2} \):

\[ P(E_2 \cap E_1) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]

### Step 2: Find \( P(E_1 \cap E_2) \)

From the previous step, we have already determined:

\[ P(E_1 \cap E_2) = \frac{1}{4} \]

### Step 3: Find \( P(E_2) \)

Using the definition of conditional probability again:

\[ P(E_1 | E_2) = \frac{P(E_1 \cap E_2)}{P(E_2)} \]

Substituting the known values:

\[ \frac{1}{4} = \frac{\frac{1}{4}}{P(E_2)} \]

Cross-multiplying gives:

\[ P(E_2) = 1 \]

### Step 4: Find \( P(E'_2) \)

The probability of the complement of \( E_2 \) is given by:

\[ P(E'_2) = 1 - P(E_2) \]

Substituting the value we found for \( P(E_2) \):

\[ P(E'_2) = 1 - 1 = 0 \]

Thus, the final answer is:

\[ P(E'_2) = 0 \]