At 100 college campuses, 1200 full-time undergraduate students were surveyed on their credit card usage. Among juniors, 65% reported that they didn't have a credit card in their own name, and 23% reported that they had at least one credit card in their own name and they paid their credit card balance in full each month.
Consider this probability experiment. A college junior is randomly selected. The junior is interviewed and then categorized into one of the following three categories: she does not have a credit card in her own name, she has at least one credit card in her own name and pays her credit card balance in full each month, or she has at least one credit card in her own name and maintains a credit card balance.
Consider the following events.
O = The junior does not have a credit card in her own name
C = The junior has at least one credit card in her own name
F = The junior has at least one credit card in her own name and pays her credit card balance in full each month
B = The junior has at least one credit card in her own name and maintains a credit card balance
1. The events O and B (are, are not) mutually exclusive.
2. The events B and F (are, are not) all-inclusive.
3. Event B (is, is not) composed of other events being considered in the experiment.
4. The set S = {O, C, F} (is, is not) the correct description of the list of all outcomes for the probability experiment because the events in S are (mutually exclusive and all-inclusive, not mutually exclusive, not all-inclusive)
5. Find P'(B) and P'(C), and list the probabilities
P'(B):
P'(C):
I am really confused on this whole problem, I would really appreciate it if someone could help me out!
yes
Sure! I can help you with that. Let's go step-by-step:
1. The events O and B are mutually exclusive if they cannot occur at the same time. In this case, if a junior does not have a credit card in her own name (event O), it implies that she cannot have a credit card balance (event B). Therefore, O and B are mutually exclusive.
2. The events B and F are all-inclusive if every outcome in event B is also contained within event F. In this case, if a junior has at least one credit card in her own name and maintains a credit card balance (event B), it does not necessarily mean that she pays her credit card balance in full each month (event F). Therefore, B and F are not all-inclusive.
3. Event B is composed of other events being considered in the experiment if all the outcomes within event B can be categorized into different events in the experiment. In this case, event B can be divided into two sub-events: F (junior pays credit card balance in full each month) and another event where the junior does not pay her credit card balance in full each month. Therefore, event B is composed of other events being considered in the experiment.
4. The set S = {O, C, F} is not the correct description of the list of all outcomes for the probability experiment because the events in S are mutually exclusive and all-inclusive.
5. To calculate P'(B) and P'(C), we need some additional information. Do you have the probabilities of event O and event C?
Once we have the probabilities for event O and event C, we can calculate P'(B) using the formula P'(B) = 1 - P(B), where P(B) is the probability of event B. Similarly, we can calculate P'(C) using the formula P'(C) = 1 - P(C).
Please provide the probabilities for event O and event C, and I'll be able to calculate P'(B) and P'(C) for you.
Sure! Let's go through each question step by step.
1. The events O and B (are, are not) mutually exclusive.
To determine if two events are mutually exclusive, we check if they can occur at the same time or not. In this case, the events O and B refer to whether a junior has a credit card in their own name but does not pay the balance (event O), or maintains a credit card balance (event B).
Since it is possible for a junior to have a credit card in their own name and maintain a balance (event B), while also not having a credit card in their own name (event O), these two events can occur at the same time. Therefore, the events O and B are NOT mutually exclusive.
2. The events B and F (are, are not) all-inclusive.
To determine if two events are all-inclusive, we need to check if one event implies the occurrence of the other. In this case, the events B and F refer to whether a junior has at least one credit card in their own name and maintains a credit card balance (event B), or they have at least one credit card in their own name and pay the balance in full each month (event F).
Since paying the credit card balance in full each month (event F) is a specific condition within maintaining a balance (event B), it means that if a junior is in category F, they are also in category B. Therefore, the event F is included within the event B. Hence, the events B and F are all-inclusive.
3. Event B (is, is not) composed of other events being considered in the experiment.
Event B refers to whether a junior has at least one credit card in their own name and maintains a credit card balance. In this case, event B is not composed of other events being considered in the experiment because it is a standalone category. It represents a specific combination of having a credit card and maintaining a balance.
4. The set S = {O, C, F} (is, is not) the correct description of the list of all outcomes for the probability experiment because the events in S are (mutually exclusive and all-inclusive, not mutually exclusive, not all-inclusive).
The set S = {O, C, F} represents the list of all outcomes for the probability experiment. We have already determined that events O and B are not mutually exclusive, so S cannot be mutually exclusive. Additionally, we have also determined that events B and F are all-inclusive, so S cannot be all-inclusive.
Therefore, the correct description is that the events in S are NOT mutually exclusive and NOT all-inclusive.
5. Find P'(B) and P'(C), and list the probabilities.
P'(B) represents the probability of event B occurring, and P'(C) represents the probability of event C occurring.
Based on the given information, we know that 23% of juniors have at least one credit card in their own name and pay their credit card balance in full each month (event F). Therefore, the probability of event B occurring is equal to the probability of event F, which is 23%. Hence, P'(B) = 0.23.
From the given information, it is not explicitly stated what percentage of juniors have at least one credit card in their own name (event C). However, we know that 65% of juniors do not have a credit card in their own name (event O). Therefore, we can subtract this percentage from 100% to find the probability of event C occurring: P'(C) = 100% - 65% = 35%.
So, P'(B) = 0.23 and P'(C) = 0.35.