A and B are two events such that
`P(A)=0.54, P(B)=0.69` and `P(AnnB)=0.35`
Find (i) `P(AuuB)` (ii) `P(A'nnB')`
(iii) `P(AnnB')` (iv)`P(BnnA')`

Answer

To solve the problem step by step, we will use the given probabilities and the fundamental rules of probability.

### Given:
- \( P(A) = 0.54 \)
- \( P(B) = 0.69 \)
- \( P(A \cap B) = 0.35 \)

### (i) Find \( P(A \cup B) \)

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 values:
\[ P(A \cup B) = 0.54 + 0.69 - 0.35 \]
Calculating:
\[ P(A \cup B) = 1.23 - 0.35 = 0.88 \]

### (ii) Find \( P(A' \cap B') \)

Using De Morgan's Law:
\[ P(A' \cap B') = P((A \cup B)') \]
So we need to find:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
Substituting the value we found in part (i):
\[ P(A' \cap B') = 1 - 0.88 = 0.12 \]

### (iii) Find \( P(A \cap B') \)

Using the formula:
\[ P(A \cap B') = P(A) - P(A \cap B) \]
Substituting the values:
\[ P(A \cap B') = 0.54 - 0.35 \]
Calculating:
\[ P(A \cap B') = 0.19 \]

### (iv) Find \( P(B \cap A') \)

Using the formula:
\[ P(B \cap A') = P(B) - P(A \cap B) \]
Substituting the values:
\[ P(B \cap A') = 0.69 - 0.35 \]
Calculating:
\[ P(B \cap A') = 0.34 \]

### Final Answers:
1. \( P(A \cup B) = 0.88 \)
2. \( P(A' \cap B') = 0.12 \)
3. \( P(A \cap B') = 0.19 \)
4. \( P(B \cap A') = 0.34 \)