Two events A and B are such that P(B) = 0.55 and P(A `nn` B') = 0.15, then probability that at least one of A, B occurs is

Options

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

Answer

To find the probability that at least one of the events A or B occurs, we need to calculate \( P(A \cup B) \). Given the information:

- \( P(B) = 0.55 \)
- \( P(A \cap B') = 0.15 \)

Where \( B' \) is the complement of event B. The intersection \( A \cap B' \) represents the probability of event A occurring while event B does not occur.

### Step-by-step Solution:

1. **Understand the Events**:
- We have two events A and B.
- \( P(B) \) is the probability of event B occurring.
- \( P(A \cap B') \) is the probability of event A occurring while event B does not.

2. **Use the Formula for Union of Two Events**:
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, we do not have \( P(A) \) or \( P(A \cap B) \) directly.

3. **Relate \( P(A) \) to Given Probabilities**:
We can express \( P(A) \) in terms of the known probabilities:
\[ P(A) = P(A \cap B) + P(A \cap B') \]
Rearranging gives us:
\[ P(A) = P(A \cap B) + 0.15 \]

4. **Express \( P(A \cap B) \)**:
Since \( P(A \cap B) \) can be found using the total probability of B:
\[ P(B) = P(A \cap B) + P(A' \cap B) \]
Here, \( P(A' \cap B) \) is the probability of B occurring while A does not. We don't have this value directly, but we can express \( P(A \cap B) \) as:
\[ P(A \cap B) = P(B) - P(A' \cap B) \]

5. **Find \( P(A \cup B) \)**:
We can now substitute \( P(A) \) in the union formula:
\[ P(A \cup B) = (P(A \cap B) + 0.15) + 0.55 - P(A \cap B) \]
Simplifying this gives:
\[ P(A \cup B) = 0.15 + 0.55 = 0.70 \]

6. **Final Result**:
Therefore, the probability that at least one of A or B occurs is:
\[ P(A \cup B) = 0.70 \]

### Conclusion:
The probability that at least one of A or B occurs is \( \boxed{0.70} \).