Two events A and B are such that P(B) = 0 . 55 and P(AB') = 0 . 15 . The probability of occurence of at least one of event is

Options

A. `0.70`
B. `0.20`
C. `0.35`
D. `0.30`

Answer

To find the probability of the occurrence of at least one of the events A or B, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- We are given \( P(B) = 0.55 \)
- We are also given \( P(A \cap B') = 0.15 \)

2. **Understand the Events**:
- \( P(A \cap B') \) represents the probability of event A occurring while event B does not occur.

3. **Use the Formula**:
- We know that the probability of event A can be expressed as:
\[ P(A) = P(A \cap B) + P(A \cap B') \]
- However, we do not have \( P(A \cap B) \) directly, but we can express \( P(A) \) in terms of \( P(B) \) and \( P(A \cap B') \).

4. **Express \( P(A) \)**:
- From the total probability, we can derive:
\[ P(A) = P(B) + P(A \cap B') \]
- This is based on the idea that the total probability of A can be split into two parts: when B occurs and when B does not occur.

5. **Substitute the Values**:
- Substitute the known probabilities into the equation:
\[ P(A) = 0.55 + 0.15 \]

6. **Calculate \( P(A) \)**:
- Adding these values gives:
\[ P(A) = 0.70 \]

7. **Find the Probability of At Least One Event**:
- The probability of at least one of the events A or B occurring is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- However, since we do not have \( P(A \cap B) \), we can use the fact that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- But since we need \( P(A \cup B) \) directly, we can also consider that:
\[ P(A \cup B) = P(B) + P(A \cap B') \]
- Therefore:
\[ P(A \cup B) = 0.55 + 0.15 = 0.70 \]

8. **Final Result**:
- Thus, the probability of occurrence of at least one of the events A or B is:
\[ P(A \cup B) = 0.70 \]

### Conclusion:
The probability of occurrence of at least one of event A or B is \( 0.70 \).