Two independent events A and B are such that `P (A uu B) = (2)/(3) and `P(A n B) = (1)/(6)`. If `P(B) < P(A), then what is `P(B)` equal to ?

Options

A. `(1)/(4)`
B. `(1)/(3)`
C. `(1)/(2)`
D. `(1)/(6)`

Explanation

To solve the problem, we need to find the probability of event B, given the probabilities of the union and intersection of two independent events A and B.

### Step-by-Step Solution:

1. **Write down the given information:**
- \( P(A \cup B) = \frac{2}{3} \)
- \( P(A \cap B) = \frac{1}{6} \)
- \( P(B) < P(A) \)

2. **Use the formula for the union of two independent events:**
The formula for the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ \frac{2}{3} = P(A) + P(B) - \frac{1}{6} \]

3. **Rearrange the equation:**
To isolate \( P(A) + P(B) \):
\[ P(A) + P(B) = \frac{2}{3} + \frac{1}{6} \]
To add these fractions, convert \( \frac{2}{3} \) to sixths:
\[ \frac{2}{3} = \frac{4}{6} \]
Therefore:
\[ P(A) + P(B) = \frac{4}{6} + \frac{1}{6} = \frac{5}{6} \]

4. **Use the independence of events:**
Since A and B are independent, we have:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Substituting the known value:
\[ \frac{1}{6} = P(A) \cdot P(B) \]

5. **Express \( P(A) \) in terms of \( P(B) \):**
From the equation \( P(A) + P(B) = \frac{5}{6} \), we can express \( P(A) \):
\[ P(A) = \frac{5}{6} - P(B) \]

6. **Substitute \( P(A) \) into the independence equation:**
Substitute \( P(A) \) into \( P(A) \cdot P(B) = \frac{1}{6} \):
\[ \left(\frac{5}{6} - P(B)\right) \cdot P(B) = \frac{1}{6} \]

7. **Expand and rearrange the equation:**
\[ \frac{5}{6}P(B) - P(B)^2 = \frac{1}{6} \]
Rearranging gives:
\[ P(B)^2 - \frac{5}{6}P(B) + \frac{1}{6} = 0 \]

8. **Multiply through by 6 to eliminate fractions:**
\[ 6P(B)^2 - 5P(B) + 1 = 0 \]

9. **Use the quadratic formula to solve for \( P(B) \):**
The quadratic formula is given by:
\[ P(B) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 6, b = -5, c = 1 \):
\[ P(B) = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 6 \cdot 1}}{2 \cdot 6} \]
\[ P(B) = \frac{5 \pm \sqrt{25 - 24}}{12} \]
\[ P(B) = \frac{5 \pm 1}{12} \]
This gives two possible solutions:
\[ P(B) = \frac{6}{12} = \frac{1}{2} \quad \text{or} \quad P(B) = \frac{4}{12} = \frac{1}{3} \]

10. **Determine the correct value of \( P(B) \):**
Since we know \( P(B) < P(A) \), we need to check which value satisfies this condition.
If \( P(B) = \frac{1}{2} \), then \( P(A) = \frac{5}{6} - \frac{1}{2} = \frac{5}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3} \). Here, \( P(B) \) is not less than \( P(A) \).

If \( P(B) = \frac{1}{3} \), then \( P(A) = \frac{5}{6} - \frac{1}{3} = \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2} \). Here, \( P(B) < P(A) \) holds true.

Thus, the final answer is:
\[ P(B) = \frac{1}{3} \]