Question
Assume that a student is chosen at random from a class. Determine whether the events A and B are independent, mutually exclusive, or neither.
A: The student is a man.
B: The student belongs to a sorority.
A: The student is a man.
B: The student belongs to a sorority.
Answers
Answered by
GPT-4o mini
To determine the relationship between events A and B, we need to analyze them based on their definitions:
- **Event A**: The student is a man.
- **Event B**: The student belongs to a sorority.
### Mutually Exclusive:
Two events are mutually exclusive if they cannot occur at the same time. In this case, a student cannot be both a man and a member of a sorority simultaneously (assuming traditional definitions of sororities as female-only organizations). Hence, events A and B are mutually exclusive.
### Independent:
Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other event. Given that a man cannot belong to a sorority, the occurrence of event A (a student being a man) directly affects the likelihood of event B (the student belonging to a sorority) because if event A occurs, event B cannot occur. Therefore, they are not independent.
### Conclusion:
Events A and B are **mutually exclusive**.
- **Event A**: The student is a man.
- **Event B**: The student belongs to a sorority.
### Mutually Exclusive:
Two events are mutually exclusive if they cannot occur at the same time. In this case, a student cannot be both a man and a member of a sorority simultaneously (assuming traditional definitions of sororities as female-only organizations). Hence, events A and B are mutually exclusive.
### Independent:
Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other event. Given that a man cannot belong to a sorority, the occurrence of event A (a student being a man) directly affects the likelihood of event B (the student belonging to a sorority) because if event A occurs, event B cannot occur. Therefore, they are not independent.
### Conclusion:
Events A and B are **mutually exclusive**.