If A and B are two events such that `P(A uu B) = (5)/(6) , P(A nn B) = (1)/(3) ` and `P(barB) = (1)/(3)` then P(A) =

Options

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

Answer

To solve the problem, we need to find \( P(A) \) given the following information:

1. \( P(A \cup B) = \frac{5}{6} \)
2. \( P(A \cap B) = \frac{1}{3} \)
3. \( P(\bar{B}) = \frac{1}{3} \)

### Step-by-Step Solution:

**Step 1: Find \( P(B) \)**

We know that the probability of an event plus the probability of its complement equals 1. Thus, we can find \( P(B) \) as follows:

\[ P(B) = 1 - P(\bar{B}) = 1 - \frac{1}{3} = \frac{2}{3} \]

**Step 2: Use the formula for the union of two events**

The formula for the probability of the union of two events \( A \) and \( B \) is given by:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

We can rearrange this formula to solve for \( P(A) \):

\[ P(A) = P(A \cup B) - P(B) + P(A \cap B) \]

**Step 3: Substitute the known values into the formula**

Now we can substitute the values we have into the rearranged formula:

\[ P(A) = P(A \cup B) - P(B) + P(A \cap B) \]

Substituting the known values:

\[ P(A) = \frac{5}{6} - \frac{2}{3} + \frac{1}{3} \]

**Step 4: Simplify the expression**

To simplify the expression, we first convert all fractions to have a common denominator. The common denominator for 6 and 3 is 6:

\[ P(A) = \frac{5}{6} - \frac{4}{6} + \frac{2}{6} \]

Now, combine the fractions:

\[ P(A) = \frac{5 - 4 + 2}{6} = \frac{3}{6} = \frac{1}{2} \]

### Final Answer:

Thus, the probability \( P(A) \) is:

\[ P(A) = \frac{1}{2} \]