If events \( A \) and \( B \) are from the same sample space, and if \( P(A)=0.8 \) and \( P(B)=0.9 \), can events \( A \) and \( B \) be mutually exclusive? Explain. Choose the correct answer below. A. All probabilities are less than or equal to 1 and greater than or equal to 0 . If events are mutually exclusive, then \( P(A \cup B)=P(A)+P(B) \). Therefore, if \( A \) and \( B \) are mutually exclusive, \( P(A \cup \) \( B)=P(A)+P(B)=0.8+0.9=1.7 \), which is impossible. Therefore, events \( A \) and \( B \) are not mutually exclusive. B. All probabilities are less than or equal to 1 and greater than or equal to 0 . If events are mutually exclusive, then \( P(A \cup B)=P(A)+P(B) \). Therefore, if \( A \) and \( B \) are mutually exclusive, \( P(A \cup B)=P(A)+P(B)=0.8+0.9=0 \), which is possible. Therefore, events \( A \) and \( B \) are mutually exclusive. C. All probabilities are less than or equal to 1 and greater than or equal to 0 . If events are mutually exclusive, then \( P(A \cup B)=P(A)+P(B) \). Therefore, if \( A \) and \( B \) are mutually exclusive, \( P(A \cup B)=P(A)+P(B)=0.8+0.9=1 \), which is possible. Therefore, events \( A \) and \( B \) are mutually exclusive. D. All probabilities are less than or equal to 1 and greater than or equal to 0 . If events are mutually exclusive, then \( P(A \cup B)=P(A)+P(B) \). Therefore, if \( A \) and \( B \) are mutually exclusive, \( P(A \cup \) \( B)=P(A)+P(B)=0.8+0.9=1.7 \), which is possible. Therefore, events \( A \) and \( B \) are mutually exclusive. D. A
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All probabilities in a sample space must be between 0 and 1, which sets a clear limit on how much probability can be distributed among mutually exclusive events. Since \( P(A) = 0.8 \) and \( P(B) = 0.9 \), if you combine these probabilities where the events are mutually exclusive (meaning they cannot happen at the same time), you arrive at \( P(A \cup B) = P(A) + P(B) = 1.7 \). This is not possible, as it exceeds 1! Now, let’s think about what mutual exclusivity really means! When two events are mutually exclusive, it implies that one event happening means the other cannot. Given that the sum of the probabilities exceeds 1, we can conclude that \( A \) and \( B \) must have some overlap; hence, they cannot be mutually exclusive. So, the correct answer is indeed A!
