E. Chapter 3 HW- Probability Topics 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 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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When you roll two dice, the concept of mutually exclusive events pops up pretty quickly! If one event's happening means the other can’t happen, then they're mutually exclusive. For example, if event \( A \) is rolling a four or five on the first die and event \( B \) is rolling an odd number on the second die with the total not exceeding eight, you can check if they overlap. With some playful calculations, you can find that they're not exclusive, which means they can occur together once in a while! Now let's chat about independence! Events are independent if knowing one doesn't change the chance of the other happening. In our dice scenario, if rolling a four or five on the first die influences the odds of rolling an odd number on the second die, then they’re dependent! You can always visualize this with a fun probability tree or table to help see the connections and results clearly, turning a complicated concept into a colorful journey of chance!
