The probability of the simultaneous occurrence of two events A and B is p. If the probability that exactly one of A, B occurs is q, then which of the following alternatives is incorrect ?

Options

A. `P(barA) + P(barB) = 2 + 2q - p`
B. `P(barA) + P(barE) = 2- 2 p - q`
C. `P(A cap B // A cup B) = (p)/(p +q)`
D. `P(barA cap barB) = 1 - p - q`

Answer

To solve the problem, we need to analyze the given probabilities and derive the relationships between them. Let's break it down step by step.

### Step 1: Understand the Given Information
We are given:
- The probability of the simultaneous occurrence of two events A and B is \( P(A \cap B) = p \).
- The probability that exactly one of A or B occurs is \( q \).

### Step 2: Express the Probability of Exactly One Event Occurring
The probability that exactly one of the events A or B occurs can be expressed as:
\[ P(A \text{ only}) + P(B \text{ only}) = P(A) + P(B) - 2P(A \cap B) \]
Thus, we can write:
\[ q = P(A) + P(B) - 2p \]

### Step 3: Rearranging the Equation
From the equation \( q = P(A) + P(B) - 2p \), we can express \( P(A) + P(B) \) in terms of \( q \) and \( p \):
\[ P(A) + P(B) = q + 2p \]

### Step 4: Consider the Complement of Events
The probability of neither A nor B occurring can be expressed as:
\[ P(A^c \cap B^c) = 1 - P(A \cup B) \]
Using the formula for the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting \( P(A \cap B) = p \):
\[ P(A \cup B) = P(A) + P(B) - p \]
Thus, we have:
\[ P(A^c \cap B^c) = 1 - (P(A) + P(B) - p) \]
Substituting \( P(A) + P(B) = q + 2p \):
\[ P(A^c \cap B^c) = 1 - (q + 2p - p) = 1 - (q + p) = 1 - q - p \]

### Step 5: Analyze the Options
Now we need to analyze the provided options to determine which one is incorrect based on our derived equations. The options would typically involve relationships between \( p \) and \( q \) based on the derived equations.

### Conclusion
To conclude, we would compare the derived equations with the options given in the question to identify the incorrect statement.