Let's go through the problem based on the image you provided. You are asked to identify whether events are Mutually Exclusive, Independent, or Neither.

Definitions:

1. Mutually Exclusive: Two events are mutually exclusive if they cannot happen at the same time. For example, flipping a coin and getting both heads and tails in the same flip is impossible.

2. Independent: Two events are independent if the occurrence of one does not affect the probability of the other. For example, flipping a coin and rolling a die are independent because the outcome of the coin does not affect the die roll.

3. Neither: If events are not mutually exclusive and not independent, they fall under this category.

Now, let's evaluate each part:

Part (a):

Event: You and a randomly selected student both earn A's in the course.

• These events are Independent because the performance of one student does not affect the performance of another student. Therefore, the correct answer is Independent.

Part (b):

Event: You and your class partner both earn A's in the course.

• While you marked this as "Independent," the system flagged it as incorrect. Likely, this could be because your class partner's performance might be somewhat related to yours (e.g., study partners could influence each other). Thus, this situation could be classified as Neither. Your answer of "Independent" was likely wrong for this reason.

Part (c):

Event: If two events can occur at the same time, they must be independent.

• The correct answer is False. Just because two events can occur at the same time doesn't mean they are independent. For instance, both rain and carrying an umbrella can occur simultaneously, but they are not independent (as carrying an umbrella might depend on the fact that it's raining).

Let me know if you have any questions, or need further explanation!

Related questions:

1. What would be an example of mutually exclusive events in this course?
2. How do we calculate the probability of two independent events both happening?
3. Can mutually exclusive events ever be independent?
4. What is the difference between "disjoint" and "independent" events in probability theory?
5. How does conditional probability relate to the concept of independence?

Tip:

When determining whether events are independent, check if the probability of one event happening changes the likelihood of the other.