If two events A and B are such that `P(A^(C))=0*3`
`P(B)=0*4 and P(AB^(C))=0*5` then
`P[B//(AcupB^(C))]=` _____

Answer

To solve the problem, we need to find \( P[B | (A^C \cup B^C)] \).

### Step 1: Understand the given probabilities
1. \( P(A^C) = 0.3 \) implies \( P(A) = 1 - P(A^C) = 1 - 0.3 = 0.7 \).
2. \( P(B) = 0.4 \).
3. \( P(A \cap B^C) = 0.5 \).

### Step 2: Find \( P(A \cap B) \)
Using the formula for the probability of the union of two events:
\[ P(A) = P(A \cap B) + P(A \cap B^C) \]
Rearranging gives us:
\[ P(A \cap B) = P(A) - P(A \cap B^C) = 0.7 - 0.5 = 0.2 \]

### Step 3: Find \( P(A^C \cup B^C) \)
Using De Morgan's law:
\[ P(A^C \cup B^C) = 1 - P(A \cap B) \]
We already found \( P(A \cap B) = 0.2 \), so:
\[ P(A^C \cup B^C) = 1 - 0.2 = 0.8 \]

### Step 4: Find \( P(B \cap (A^C \cup B^C)) \)
Using the formula:
\[ P(B \cap (A^C \cup B^C)) = P(B) - P(B \cap A) \]
We know \( P(B) = 0.4 \) and \( P(B \cap A) = 0.2 \):
\[ P(B \cap (A^C \cup B^C)) = 0.4 - 0.2 = 0.2 \]

### Step 5: Calculate \( P[B | (A^C \cup B^C)] \)
Using the conditional probability formula:
\[ P[B | (A^C \cup B^C)] = \frac{P(B \cap (A^C \cup B^C))}{P(A^C \cup B^C)} \]
Substituting the values we found:
\[ P[B | (A^C \cup B^C)] = \frac{0.2}{0.8} = \frac{1}{4} \]

### Final Answer
Thus, the value of \( P[B | (A^C \cup B^C)] \) is \( \frac{1}{4} \).

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