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If 0 lt P(A) lt 1, 0 lt P(B) lt 1 and P...

If ` 0 lt P(A) lt 1, 0 lt P(B) lt 1` and `P(A cup B)=P(A)+P(B)-P(A)P(B)`, then

A

`P(B-A)=P(A)-P(A)`

B

`P(A'cupB')=P(A')+P(B')`

C

`P((AcupB)')=P(A')P(B')`

D

`P((A)/(B))=P(A)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we start with the given equation: \[ P(A \cup B) = P(A) + P(B) - P(A)P(B) \] This equation is a standard result in probability theory, which holds true when events A and B are independent. We need to analyze the implications of this equation. ### Step 1: Understanding the Given Equation The equation provided states that the probability of the union of two events A and B is equal to the sum of their individual probabilities minus the product of their probabilities. This is a characteristic of independent events. ### Step 2: Rearranging the Equation We can rearrange the equation to express it in a different form: \[ P(A \cup B) + P(A)P(B) = P(A) + P(B) \] ### Step 3: Analyzing the Implications From the rearranged equation, we can see that if we denote \( P(A \cup B) \) as \( x \), we have: \[ x + P(A)P(B) = P(A) + P(B) \] This implies that: \[ x = P(A) + P(B) - P(A)P(B) \] ### Step 4: Considering the Complement Now, let's consider the complements of A and B. The complement of the union of A and B is given by: \[ P(A' \cap B') = 1 - P(A \cup B) \] Using the previous steps, we can express this as: \[ P(A' \cap B') = 1 - (P(A) + P(B) - P(A)P(B)) \] ### Step 5: Final Expression Thus, we can conclude that: \[ P(A' \cap B') = 1 - P(A) - P(B) + P(A)P(B) \] This shows the relationship between the probabilities of the events and their complements. ### Conclusion The given equation \( P(A \cup B) = P(A) + P(B) - P(A)P(B) \) confirms that A and B are independent events.
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