In parts (a) and (b), identify whether the events are Disjoint (mutually exclusive), independent,
or neither (events cannot be both disjoint and independent).
a) You and a randomly selected student from your class both earn A’s in this course.
independent
neither
disjoint
b) You and your class partner both earn A’s in this course.
independent
neither
c) If two events can occur at the same time, they must be independent.
false
true

Question

UpStudy AI · Accepted Answer

**(a)** Let  
$$ A = $$ "You earn an A"  
$$ B = $$ "A randomly selected student earns an A"

Assuming that each student’s grade is determined independently, the event that you earn an A does not affect the probability that a randomly chosen student earns an A. Thus, the events are independent in this context.

**(b)** Let  
$$ A = $$ "You earn an A"  
$$ B = $$ "Your class partner earns an A"

Because you and your class partner likely have similar abilities, work together, or are affected by similar factors (e.g., the same instructor, possible collaboration), the event that you earn an A can affect the probability that your partner earns an A. Therefore, the events are neither independent nor disjoint (since both outcomes can occur simultaneously).

**(c)** The statement claims:  
"If two events can occur at the same time, they must be independent."

This is not true. Two events occurring at the same time (i.e., they are not mutually exclusive) does not imply that the occurrence of one does not affect the probability of occurrence of the other. Independence requires that  
$$
P(A \cap B) = P(A) \cdot P(B),
$$  
which is not guaranteed simply because the events can occur simultaneously.

Thus, the answer is false.
**(a)** Independent **(b)** Neither **(c)** False

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