Given two events A and B. If odds against A are 2 : 1 and those in fovour of `AcupB` are as 3 : 1, then

Options

A. `1/2leP(B)le3/9`
B. `5/12leP(B)le3/4`
C. `1/2leP(B)le3/5`
D. `1/3leP(B)le3/4`

Answer

To solve the problem, we need to find the probability of event B given the odds against event A and the odds in favor of the union of events A and B. Let's break it down step by step.

### Step 1: Understand the odds against A
The odds against event A are given as 2:1. This means that for every 2 occurrences of A not happening, A happens once.

- The probability of A not happening (A complement) can be calculated as:
\[ P(A') = \frac{2}{2 + 1} = \frac{2}{3} \]

- Therefore, the probability of event A happening is:
\[ P(A) = 1 - P(A') = 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 2: Understand the odds in favor of A ∪ B
The odds in favor of A ∪ B are given as 3:1. This means that for every 3 occurrences of A ∪ B happening, there is 1 occurrence of it not happening.

- The probability of A ∪ B can be calculated as:
\[ P(A ∪ B) = \frac{3}{3 + 1} = \frac{3}{4} \]

### Step 3: Use the formula for the probability of the union of two events
We know that:
\[ P(A ∪ B) = P(A) + P(B) - P(A ∩ B) \]

From the above, we can rearrange it to find P(B):
\[ P(B) = P(A ∪ B) - P(A) + P(A ∩ B) \]

### Step 4: Establish inequalities for P(B)
Since \( P(A ∩ B) \) cannot be negative, we can establish the following inequality:
\[ P(A ∪ B) \leq P(A) + P(B) \]
Substituting the known values:
\[ \frac{3}{4} \leq \frac{1}{3} + P(B) \]
This simplifies to:
\[ P(B) \geq \frac{3}{4} - \frac{1}{3} \]
Finding a common denominator (12):
\[ P(B) \geq \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \]

### Step 5: Establish the upper limit for P(B)
Since \( P(B) \) must also be less than or equal to \( P(A ∪ B) \):
\[ P(B) \leq P(A ∪ B) = \frac{3}{4} \]

### Step 6: Combine the inequalities
Now we have:
\[ \frac{5}{12} \leq P(B) \leq \frac{3}{4} \]

### Conclusion
Thus, the probability of event B lies between \(\frac{5}{12}\) and \(\frac{3}{4}\).