Suppose that we want to find the probability that at least 40 people in a class have done their homework. Using set notation describe this event E. Then using set notation find the complement.

Let's define the event \( E \) as follows:

1. **Event \( E \):** The event that at least 40 people in the class have done their homework. Using set notation, this can be written as:
   \[
   E = \{ X \geq 40 \}
   \]
   where \( X \) is the number of people in the class who have done their homework.

2. **Complement of Event \( E \):** The complement of \( E \), denoted as \( E^c \), is the event that fewer than 40 people have done their homework. Using set notation, this can be written as:
   \[
   E^c = \{ X < 40 \}
   \]
   This represents the scenario where less than 40 people have completed their homework.

Would you like to explore further details or calculations regarding this problem?

Here are 5 related questions for further exploration:

1. How would you calculate the probability of \( E \) given a specific distribution (e.g., binomial or normal)?
2. What assumptions must be made to model the probability of \( X \) being at least 40?
3. How does the complement rule relate to the probability of event \( E \)?
4. What are common scenarios where the complement of an event is useful in probability theory?
5. Can you describe how to calculate the expected number of students who did their homework?

**Tip:** When solving probability problems, always check if the complement of an event simplifies the calculation, as it can often be easier to compute.

Explore how to find the probability that at least 40 students in a class have completed their homework using set notation. This solution explains the event and its complement, along with key concepts like the complement rule in probability. Suitable for advanced high school students (Grades 11-12).

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