Events A and B are mutually exclusive exhaustive . If `P(E_1) =2/3 P(E_2)` , then the odds in favor of `E_2` are (i) `3 : 1` (ii) `1:3` (iii) `3:2` (iv) `2:3`

Options

A. `3 : 1`
B. `1:3`
C. `3:2`
D. `2:3`

Answer

To solve the problem step by step, we need to find the odds in favor of event \( E_2 \) given that \( P(E_1) = \frac{2}{3} P(E_2) \) and that events \( A \) and \( B \) are mutually exclusive and exhaustive.

### Step 1: Understand the relationship between probabilities
Since events \( A \) and \( B \) are mutually exclusive and exhaustive, the sum of their probabilities is equal to 1:
\[ P(E_1) + P(E_2) = 1 \]

### Step 2: Substitute the given relationship
We know from the problem statement that:
\[ P(E_1) = \frac{2}{3} P(E_2) \]
Substituting this into the equation from Step 1 gives:
\[ \frac{2}{3} P(E_2) + P(E_2) = 1 \]

### Step 3: Combine the terms
Now, we can combine the terms on the left side:
\[ \frac{2}{3} P(E_2) + \frac{3}{3} P(E_2) = 1 \]
This simplifies to:
\[ \frac{5}{3} P(E_2) = 1 \]

### Step 4: Solve for \( P(E_2) \)
To find \( P(E_2) \), we multiply both sides by \( \frac{3}{5} \):
\[ P(E_2) = \frac{3}{5} \]

### Step 5: Find \( P(E_1) \)
Now that we have \( P(E_2) \), we can find \( P(E_1) \) using the relationship we established:
\[ P(E_1) = \frac{2}{3} P(E_2) = \frac{2}{3} \times \frac{3}{5} = \frac{2}{5} \]

### Step 6: Calculate the odds in favor of \( E_2 \)
The odds in favor of an event is given by the ratio of the probability of the event occurring to the probability of it not occurring. Thus, the odds in favor of \( E_2 \) is:
\[ \text{Odds in favor of } E_2 = \frac{P(E_2)}{P(E_1)} = \frac{\frac{3}{5}}{\frac{2}{5}} = \frac{3}{2} \]

### Conclusion
Thus, the odds in favor of \( E_2 \) are \( 3:2 \).

### Final Answer
The correct option is (iii) \( 3:2 \).

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