A and B are two events such that A' and B' are mutually exclusive. If P(A) = 0.5 and P(B) = 0.6. then what is the value of `P(A//B)`?

Options

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

Answer

To solve the problem, we need to find the conditional probability \( P(A|B) \). We know that:

1. \( P(A) = 0.5 \)
2. \( P(B) = 0.6 \)
3. \( A' \) (complement of A) and \( B' \) (complement of B) are mutually exclusive.

### Step-by-Step Solution:

**Step 1: Understand the relationship between events.**
Since \( A' \) and \( B' \) are mutually exclusive, it means that the occurrence of one event prevents the occurrence of the other. This implies that \( P(A' \cap B') = 0 \).

**Hint:** Recall that mutually exclusive events cannot happen at the same time.

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**Step 2: Use De Morgan's Law.**
According to De Morgan's Law:
\[ A' \cap B' = (A \cup B)' \]
This means that the probability of the union of A and B is:
\[ P(A \cup B) = 1 - P(A' \cap B') = 1 - 0 = 1 \]

**Hint:** Remember that the total probability of all outcomes is 1.

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**Step 3: Apply the formula for the union of two events.**
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ 1 = P(A) + P(B) - P(A \cap B) \]
\[ 1 = 0.5 + 0.6 - P(A \cap B) \]

**Hint:** This formula helps us relate the probabilities of individual events to their intersection.

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**Step 4: Solve for \( P(A \cap B) \).**
Rearranging the equation gives us:
\[ P(A \cap B) = 0.5 + 0.6 - 1 = 0.1 \]

**Hint:** Isolate the term you want to find by moving other terms to the opposite side of the equation.

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**Step 5: Calculate the conditional probability \( P(A|B) \).**
The formula for conditional probability is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Substituting the values we have:
\[ P(A|B) = \frac{0.1}{0.6} = \frac{1}{6} \]

**Hint:** Conditional probability gives you the probability of event A occurring given that event B has occurred.

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### Final Answer:
The value of \( P(A|B) \) is \( \frac{1}{6} \).