e. Are \( A \) and \( B \) mutually exclusive events? It is possible to have a four or a five on the first die rolling followed by an odd number on the second die when the sum of these two numbers is at A most eight. Numerically, we have \( P(A \) AND \( B)=\frac{1}{9} \), so the events are not mutually exclusive. It is impossible to have a four or a five on the first die rolling followed by an odd number on the second die when the sum of these two numbers is B at most eight. Numerically, we have \( P(A \) AND \( B)=0 \), so the events are mutually exclusive. f. Are \( A \) and \( B \) independent events? Two events \( A \) and \( B \) are not independent because the occurrence of one affects the chance the other occurs. Numerically, \( P(A \mid B) \neq P(A) \), so events \( A \) and \( B \) are not independent. Two events \( A \) and \( B \) are independent because the occurrence of one does not affect the chance the other occurs. Numerically, B \( P(A \mid B)=P(A) \), so events \( A \) and \( B \) are independent.
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Beyond the Answer
In probability theory, mutually exclusive events cannot occur at the same time. For example, when rolling two dice, if one event is getting a four on the first die (Event A), and another is getting an odd number on the second die (Event B), they are considered mutually exclusive if they cannot both happen together under the given condition (like the sum being at most eight). Checking the conditions will give you insights into how these events interact! When it comes to independence, this concept refers to whether the occurrence of one event affects the occurrence of another. If knowing that Event A occurs changes the probability of Event B occurring (i.e., \( P(A \mid B) \neq P(A) \)), then they are not independent. Always visualize these with fun experiments like rolling two dice together and observing the outcomes to see how they correlate or don't!
