Let A and B be two events of random experiment such that P(A') = 0.3, P(B) = 0.4 and P( A`nn` B' ) = 0.5, then P(A `uu`B) + P(B|A `uu`B') = ____

Answer

To solve the problem step by step, we will use the given probabilities and apply the relevant formulas from probability theory.

### Step 1: Find P(A)
We know that:
\[ P(A') = 0.3 \]
Using the complement rule:
\[ P(A) = 1 - P(A') = 1 - 0.3 = 0.7 \]

**Hint:** Remember that the probability of an event and its complement always sum to 1.

### Step 2: Identify Given Probabilities
We have:
- \( P(A) = 0.7 \)
- \( P(B) = 0.4 \)
- \( P(A' \cap B') = 0.5 \)

### Step 3: Find P(A ∩ B)
We can use the relationship:
\[ P(A) = P(A \cap B) + P(A \cap B') \]
We know that:
\[ P(A \cap B') = P(A') - P(A' \cap B) \]
Since \( P(A' \cap B) = P(B) - P(A \cap B) \), we can rewrite it as:
\[ P(A \cap B') = P(A) - P(A \cap B) \]

From the information given:
\[ P(A' \cap B') = 0.5 \]
This means:
\[ P(A \cup B) = 1 - P(A' \cap B') = 1 - 0.5 = 0.5 \]

### Step 4: Use the Union Formula
Using the formula for the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ 0.5 = 0.7 + 0.4 - P(A \cap B) \]
This simplifies to:
\[ P(A \cap B) = 0.7 + 0.4 - 0.5 = 0.6 \]

### Step 5: Find P(A ∩ B)
Now we can find \( P(A \cap B) \):
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]
Substituting the values:
\[ P(A \cap B) = 0.7 + 0.4 - 0.5 = 0.6 \]

### Step 6: Find P(A ∪ B)
We already have:
\[ P(A \cup B) = 0.5 \]

### Step 7: Find P(B | A ∪ B')
Using the conditional probability formula:
\[ P(B | A \cup B') = \frac{P(B \cap (A \cup B'))}{P(A \cup B')} \]
We know:
\[ P(A \cup B') = 1 - P(A' \cap B) = 1 - 0.5 = 0.5 \]
Now we need to find \( P(B \cap (A \cup B')) \):
\[ P(B \cap (A \cup B')) = P(B) - P(B \cap A) = 0.4 - 0.2 = 0.2 \]

### Step 8: Calculate P(B | A ∪ B')
Substituting into the conditional probability formula:
\[ P(B | A \cup B') = \frac{0.2}{0.5} = 0.4 \]

### Step 9: Find the Final Result
Now we can find:
\[ P(A \cup B) + P(B | A \cup B') = 0.5 + 0.4 = 0.9 \]

### Final Answer
The final answer is:
\[ P(A \cup B) + P(B | A \cup B') = 0.9 \]

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