If `A` and `B` are any two events such that `P(A)=2/5` and `P(AnnB)=3/20` then the conditional probability `P(A|(A'uuB'))` where `A'` denotes the complement of `A` is equal to

Options

A. (a) `(11)/(20)`
B. (b) `(5)/(17)`
C. (c) `(8)/(17)`
D. (d) `(1)/(4)`

Answer

To solve the problem, we need to find the conditional probability \( P(A | (A' \cup B')) \). Given that \( P(A) = \frac{2}{5} \) and \( P(A \cap B) = \frac{3}{20} \), we can follow these steps:

### Step 1: Find \( P(A' \cup B') \)

Using De Morgan's law, we know that:

\[ P(A' \cup B') = 1 - P(A \cap B) \]

Substituting the value of \( P(A \cap B) \):

\[ P(A' \cup B') = 1 - \frac{3}{20} \]

Calculating this gives:

\[ P(A' \cup B') = \frac{20}{20} - \frac{3}{20} = \frac{17}{20} \]

### Step 2: Find \( P(A \cap (A' \cup B')) \)

Using the property of probabilities, we can express this as:

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

Substituting the known values:

\[ P(A \cap (A' \cup B')) = \frac{2}{5} - \frac{3}{20} \]

To perform this subtraction, we need a common denominator. The least common multiple of 5 and 20 is 20:

\[ P(A \cap (A' \cup B')) = \frac{8}{20} - \frac{3}{20} = \frac{5}{20} \]

### Step 3: Calculate the conditional probability \( P(A | (A' \cup B')) \)

Using the formula for conditional probability:

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

Substituting the values we found:

\[ P(A | (A' \cup B')) = \frac{\frac{5}{20}}{\frac{17}{20}} \]

The \( \frac{20}{20} \) cancels out:

\[ P(A | (A' \cup B')) = \frac{5}{17} \]

### Conclusion

Thus, the conditional probability \( P(A | (A' \cup B')) \) is:

\[ \boxed{\frac{5}{17}} \]