A and B are two independent events such that their probabilities are `(3)/(10) " and " (2)/(5)` respectively. The probability of exactly one of the events happening, is

Options

A. `23//50`
B. `1//2`
C. `31//50`
D. none of these

Answer

To solve the problem, we need to find the probability of exactly one of the events A or B happening. Let's break it down step by step.

### Step-by-Step Solution:

1. **Identify the probabilities of events A and B:**
- Given \( P(A) = \frac{3}{10} \)
- Given \( P(B) = \frac{2}{5} \)

2. **Calculate the probabilities of the complements of A and B:**
- The probability that event A does not occur (denoted as \( P(A') \)):
\[ P(A') = 1 - P(A) = 1 - \frac{3}{10} = \frac{7}{10} \]
- The probability that event B does not occur (denoted as \( P(B') \)):
\[ P(B') = 1 - P(B) = 1 - \frac{2}{5} = \frac{3}{5} \]

3. **Find the probability of exactly one event happening:**
- The probability of exactly one of the events A or B happening can be calculated using the formula:
\[ P(\text{exactly one of A or B}) = P(A \cap B') + P(B \cap A') \]
- Where:
- \( P(A \cap B') = P(A) \cdot P(B') \)
- \( P(B \cap A') = P(B) \cdot P(A') \)

4. **Calculate \( P(A \cap B') \):**
\[ P(A \cap B') = P(A) \cdot P(B') = \frac{3}{10} \cdot \frac{3}{5} = \frac{9}{50} \]

5. **Calculate \( P(B \cap A') \):**
\[ P(B \cap A') = P(B) \cdot P(A') = \frac{2}{5} \cdot \frac{7}{10} = \frac{14}{50} \]

6. **Add the probabilities of the two scenarios:**
\[ P(\text{exactly one of A or B}) = P(A \cap B') + P(B \cap A') = \frac{9}{50} + \frac{14}{50} = \frac{23}{50} \]

### Final Answer:
The probability of exactly one of the events A or B happening is \( \frac{23}{50} \).