If `P(A) = (6)/(11), P(B) = (5)/(11)` and probability of at least one of the event `= (7)/(11)`, then `P((B)/(A))` is A) `(6)/(11)` B)`(4)/(5)` C)`(2)/(3)` D) 1

Options

A. `(6)/(11)`
B. `(4)/(5)`
C. `(2)/(3)`
D. 1

Answer

To solve the problem, we need to find the probability of event B given event A, denoted as \( P(B|A) \). We are given the following probabilities:

- \( P(A) = \frac{6}{11} \)
- \( P(B) = \frac{5}{11} \)
- \( P(A \cup B) = \frac{7}{11} \)

### Step 1: Find \( P(A \cap B) \)

We can use the formula for the probability of the union of two events:

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

Substituting the known values into the formula:

\[ \frac{7}{11} = \frac{6}{11} + \frac{5}{11} - P(A \cap B) \]

### Step 2: Simplify the equation

Combine the probabilities on the right side:

\[ \frac{7}{11} = \frac{11}{11} - P(A \cap B) \]

### Step 3: Solve for \( P(A \cap B) \)

Rearranging the equation gives:

\[ P(A \cap B) = \frac{11}{11} - \frac{7}{11} = \frac{4}{11} \]

### Step 4: Find \( P(B|A) \)

Now we can find \( P(B|A) \) using the definition of conditional probability:

\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

Substituting the values we found:

\[ P(B|A) = \frac{\frac{4}{11}}{\frac{6}{11}} \]

### Step 5: Simplify the fraction

When dividing fractions, we multiply by the reciprocal:

\[ P(B|A) = \frac{4}{11} \times \frac{11}{6} = \frac{4}{6} \]

### Step 6: Reduce the fraction

Now simplify \( \frac{4}{6} \):

\[ P(B|A) = \frac{2}{3} \]

### Conclusion

Thus, the probability \( P(B|A) \) is \( \frac{2}{3} \), which corresponds to option C.

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