For three events A,B and C
P (exactly one of the events A or B occurs) = P(exactly one of the events B or C occurs) = P(exactly one of the events C ir A occurs) = P and P(all the three events occur simultaneously) `=P^(2)`, where `0ltplt(1)/(2)` Then the probability of at least one of the three events A,B and C occuring is :

Options

A. `(3p+2p^(2))/(2)`
B. `(p+3p^(2))/(4)`
C. `(p+3p^(2))/(2)`
D. `(3p+3p^(2))/(4)`

Explanation

To solve the problem, we need to find the probability of at least one of the three events A, B, and C occurring, given the conditions specified in the question.

### Step-by-Step Solution:

1. **Understanding the Given Probabilities**:
We are given that:
- \( P(\text{exactly one of A or B occurs}) = P \)
- \( P(\text{exactly one of B or C occurs}) = P \)
- \( P(\text{exactly one of C or A occurs}) = P \)
- \( P(\text{all three events occur simultaneously}) = P^2 \)

2. **Expressing the Probabilities Mathematically**:
The probability of exactly one of two events occurring can be expressed as:
\[ P(A \text{ and not } B) + P(B \text{ and not } A) = P(A) + P(B) - 2P(A \cap B) \]
Therefore, we can write:
- For A and B:
\[ P(A) + P(B) - 2P(A \cap B) = P \]
- For B and C:
\[ P(B) + P(C) - 2P(B \cap C) = P \]
- For C and A:
\[ P(C) + P(A) - 2P(C \cap A) = P \]

3. **Setting Up the Equations**:
We can summarize these equations:
\[ P(A) + P(B) - 2P(A \cap B) = P \tag{1} \]
\[ P(B) + P(C) - 2P(B \cap C) = P \tag{2} \]
\[ P(C) + P(A) - 2P(C \cap A) = P \tag{3} \]

4. **Adding the Equations**:
Adding equations (1), (2), and (3):
\[ (P(A) + P(B) - 2P(A \cap B)) + (P(B) + P(C) - 2P(B \cap C)) + (P(C) + P(A) - 2P(C \cap A)) = 3P \]
This simplifies to:
\[ 2(P(A) + P(B) + P(C)) - 2(P(A \cap B) + P(B \cap C) + P(C \cap A)) = 3P \]
Dividing everything by 2:
\[ P(A) + P(B) + P(C) - (P(A \cap B) + P(B \cap C) + P(C \cap A)) = \frac{3P}{2} \tag{4} \]

5. **Finding the Probability of At Least One Event**:
The probability of at least one of the events A, B, or C occurring is given by:
\[ P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C) \]
Substituting from equation (4):
\[ P(A \cup B \cup C) = \frac{3P}{2} + P^2 \]

6. **Final Expression**:
Thus, the probability of at least one of the events A, B, or C occurring is:
\[ P(A \cup B \cup C) = \frac{3P + 2P^2}{2} \]