The probability of occurrence of an event A is `5/9`. The probability of non occurrence of the event B is `5/11`. The probability that at least one of them will occur.

Options

A. `6/11`
B. `5/9`
C. `4/9`
D. `0.8`

Answer

To solve the problem, we need to find the probability that at least one of the events A or B occurs. We can use the formula for the probability of at least one event occurring, which is:

\[ P(A \cup B) = 1 - P(A' \cap B') \]

Where:
- \( P(A \cup B) \) is the probability that at least one of the events A or B occurs.
- \( P(A') \) is the probability that event A does not occur.
- \( P(B') \) is the probability that event B does not occur.
- \( P(A' \cap B') \) is the probability that neither event A nor event B occurs.

### Step-by-Step Solution:

1. **Find the probability of event A occurring:**
\[ P(A) = \frac{5}{9} \]

2. **Calculate the probability of event A not occurring:**
\[ P(A') = 1 - P(A) = 1 - \frac{5}{9} = \frac{4}{9} \]

3. **Find the probability of event B not occurring:**
\[ P(B') = \frac{5}{11} \]

4. **Calculate the probability that neither event A nor event B occurs:**
Since A and B are independent events, we can multiply their probabilities of non-occurrence:
\[ P(A' \cap B') = P(A') \times P(B') = \frac{4}{9} \times \frac{5}{11} = \frac{20}{99} \]

5. **Calculate the probability that at least one of the events A or B occurs:**
\[ P(A \cup B) = 1 - P(A' \cap B') = 1 - \frac{20}{99} = \frac{99 - 20}{99} = \frac{79}{99} \]

6. **Final Result:**
The probability that at least one of the events A or B occurs is:
\[ P(A \cup B) = \frac{79}{99} \]