Fill in the blanks in following table:
`{:(" "P(A)" "P(B)" "P(AnnB)" "P(AuuB)),((i)" "1/3" "1/5" "1/15" "….),((ii) " "0.35" " ..." "0.25" "0.6),((iii)" "0.5" "0.35" "..." "0.7):}`

Answer

To complete the table, we will use the formula for the probability of the union of two events:

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

We will apply this formula step by step for each part of the table.

### Part (i)
Given:
- \( P(A) = \frac{1}{3} \)
- \( P(B) = \frac{1}{5} \)
- \( P(A \cap B) = \frac{1}{15} \)

We need to find \( P(A \cup B) \).

1. Convert \( P(A) \) and \( P(B) \) to a common denominator:
- The least common multiple of 3, 5, and 15 is 15.
- \( P(A) = \frac{1}{3} = \frac{5}{15} \)
- \( P(B) = \frac{1}{5} = \frac{3}{15} \)

2. Substitute the values into the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
\[ P(A \cup B) = \frac{5}{15} + \frac{3}{15} - \frac{1}{15} \]

3. Calculate:
\[ P(A \cup B) = \frac{5 + 3 - 1}{15} = \frac{7}{15} \]

### Part (ii)
Given:
- \( P(A) = 0.35 \)
- \( P(A \cap B) = 0.25 \)
- \( P(A \cup B) = 0.6 \)

We need to find \( P(B) \).

1. Substitute the values into the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
\[ 0.6 = 0.35 + P(B) - 0.25 \]

2. Simplify:
\[ 0.6 = 0.10 + P(B) \]

3. Solve for \( P(B) \):
\[ P(B) = 0.6 - 0.10 = 0.5 \]

### Part (iii)
Given:
- \( P(A) = 0.5 \)
- \( P(B) = 0.35 \)
- \( P(A \cup B) = 0.7 \)

We need to find \( P(A \cap B) \).

1. Substitute the values into the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
\[ 0.7 = 0.5 + 0.35 - P(A \cap B) \]

2. Simplify:
\[ 0.7 = 0.85 - P(A \cap B) \]

3. Solve for \( P(A \cap B) \):
\[ P(A \cap B) = 0.85 - 0.7 = 0.15 \]

### Final Table
Now we can fill in the table as follows:

| Part | P(A) | P(B) | P(A ∩ B) | P(A ∪ B) |
|------|-------|-------|----------|----------|
| (i) | 1/3 | 1/5 | 1/15 | 7/15 |
| (ii) | 0.35 | 0.5 | 0.25 | 0.6 |
| (iii)| 0.5 | 0.35 | 0.15 | 0.7 |