Questions: You are given the following information about the events A, B, and C. - P(A)=0.55 - P(B)=0.45 - P(C)=0.40 - P(A and B)=0.2475 - P(B and C)=0.1644 - P(A and C)=0.2034 Determine which (if any) pairs of the three events are independent.

Solution Steps

To determine if pairs of events are independent, we need to check if the probability of the intersection of two events equals the product of their individual probabilities. Specifically, for events \(A\) and \(B\) to be independent, \(P(A \cap B) = P(A) \cdot P(B)\). We will perform similar checks for the pairs \( (A, C) \) and \( (B, C) \).

To determine if events \(A\) and \(B\) are independent, we check the condition: \[ P(A \cap B) = P(A) \cdot P(B) \] Calculating the right side: \[ P(A) \cdot P(B) = 0.55 \cdot 0.45 = 0.2475 \] Since \(P(A \cap B) = 0.2475\), we find that: \[ P(A \cap B) = P(A) \cdot P(B) \implies \text{Events } A \text{ and } B \text{ are not independent.} \]

Next, we check if events \(B\) and \(C\) are independent: \[ P(B \cap C) = P(B) \cdot P(C) \] Calculating the right side: \[ P(B) \cdot P(C) = 0.45 \cdot 0.40 = 0.18 \] Since \(P(B \cap C) = 0.1644\), we find that: \[ P(B \cap C) \neq P(B) \cdot P(C) \implies \text{Events } B \text{ and } C \text{ are not independent.} \]

Finally, we check if events \(A\) and \(C\) are independent: \[ P(A \cap C) = P(A) \cdot P(C) \] Calculating the right side: \[ P(A) \cdot P(C) = 0.55 \cdot 0.40 = 0.22 \] Since \(P(A \cap C) = 0.2034\), we find that: \[ P(A \cap C) \neq P(A) \cdot P(C) \implies \text{Events } A \text{ and } C \text{ are not independent.} \]

Final Answer