If `P (B) = (3)/(5) , P (A//B) = (1)/(2)` and `P (A cup B) = (4)/(5)` , then `P(A cup B)' + P( A' cup B)` is equal to :

Options

A. `1//5`
B. `4//5`
C. `1//2`
D. `1`

Answer

To solve the problem, we need to find \( P(A' \cup B) + P(A \cup B') \). We will use the given probabilities to find the required values step by step.

### Step 1: Understand the given probabilities
We have:
- \( P(B) = \frac{3}{5} \)
- \( P(A|B) = \frac{1}{2} \)
- \( P(A \cup B) = \frac{4}{5} \)

### Step 2: Find \( P(A \cap B) \)
Using the definition of conditional probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Substituting the known values:
\[ \frac{1}{2} = \frac{P(A \cap B)}{\frac{3}{5}} \]
Cross-multiplying gives:
\[ P(A \cap B) = \frac{1}{2} \cdot \frac{3}{5} = \frac{3}{10} \]

### Step 3: Find \( P(A) \)
Using 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:
\[ \frac{4}{5} = P(A) + \frac{3}{5} - \frac{3}{10} \]
To solve for \( P(A) \), first convert \( \frac{3}{5} \) to tenths:
\[ \frac{3}{5} = \frac{6}{10} \]
Now substituting:
\[ \frac{4}{5} = P(A) + \frac{6}{10} - \frac{3}{10} \]
This simplifies to:
\[ \frac{4}{5} = P(A) + \frac{3}{10} \]
Convert \( \frac{4}{5} \) to tenths:
\[ \frac{4}{5} = \frac{8}{10} \]
Now we have:
\[ \frac{8}{10} = P(A) + \frac{3}{10} \]
Subtract \( \frac{3}{10} \) from both sides:
\[ P(A) = \frac{8}{10} - \frac{3}{10} = \frac{5}{10} = \frac{1}{2} \]

### Step 4: Find \( P(A' \cup B) \)
Using the formula:
\[ P(A' \cup B) = 1 - P(A \cap B') \]
We know:
\[ P(A \cap B') = P(B) - P(A \cap B) \]
Substituting the known values:
\[ P(A \cap B') = \frac{3}{5} - \frac{3}{10} \]
Convert \( \frac{3}{5} \) to tenths:
\[ P(A \cap B') = \frac{6}{10} - \frac{3}{10} = \frac{3}{10} \]
Now substituting back:
\[ P(A' \cup B) = 1 - P(A \cap B') = 1 - \frac{3}{10} = \frac{7}{10} \]

### Step 5: Find \( P(A \cup B') \)
Using the formula:
\[ P(A \cup B') = P(A) + P(B') - P(A \cap B') \]
We know:
\[ P(B') = 1 - P(B) = 1 - \frac{3}{5} = \frac{2}{5} \]
Now substituting:
\[ P(A \cup B') = P(A) + P(B') - P(A \cap B') \]
Substituting the known values:
\[ P(A \cup B') = \frac{1}{2} + \frac{2}{5} - \frac{3}{10} \]
Convert \( \frac{1}{2} \) to tenths:
\[ P(A \cup B') = \frac{5}{10} + \frac{4}{10} - \frac{3}{10} = \frac{6}{10} = \frac{3}{5} \]

### Step 6: Combine the results
Now we can find:
\[ P(A' \cup B) + P(A \cup B') = \frac{7}{10} + \frac{3}{5} \]
Convert \( \frac{3}{5} \) to tenths:
\[ \frac{3}{5} = \frac{6}{10} \]
Thus:
\[ P(A' \cup B) + P(A \cup B') = \frac{7}{10} + \frac{6}{10} = \frac{13}{10} = 1.3 \]

### Final Answer
\[ P(A' \cup B) + P(A \cup B') = 1.3 \]