The teacher of a mathematics class has written up the final exam, but wants
the questions to be random for each student. There are 28 students in the class, and
the test has 100 questions. Each question has four possible answers.
a. After reading carefully reading the scenario, determine the number of different
final exams possible. Besides your method, are there other methods to determine
the number of outcomes?
b. For this scenario, which will work better to determine the order of the questions,
permutation or combination? Justify your answer.
c. If a student doesn’t show up to take the test, how does that affect the number of
outcomes?
d. Suppose the teacher doesn’t end up randomizing the questions on the exam. Does
this affect the number of outcomes?
e. If each question on the test had the same answer (let’s say C), what’s the probability
that students taking the test will get all the questions right? Justify your answer.
Please see the answers for the Related Questions below.
a.
There are 100! final exams possible, by permuting all 100 questions.
b. see part a.
c. The experiment is not clearly defined, so neither is the number of outcomes.
If the experiment is to determine the number of exams possible, the number of outcomes does not depend on the number of students.
If the experiments is the results of the test, then yes, the number of students affects the number of outcomes (from 28 to 27)
d. There will still be 28 outcomes.
e. It depends on how much students have studied. Quesion did not say the students answer at random.
a. To determine the number of different final exams possible, we need to calculate the total number of possible combinations for the questions and their answers.
Each question has four possible answers, so for each question, there are 4 options. Since there are 100 questions on the test, the total number of outcomes can be calculated by raising 4 to the power of 100:
Total number of outcomes = 4^100 = 1.6 x 10^60
There are no other methods to determine the number of outcomes in this specific scenario because each student could potentially have a different combination of questions and answers.
b. In this scenario, permutation would be more appropriate to determine the order of questions. Permutation is used when the order of the elements matters. Since the teacher wants the questions to be random for each student, the order of the questions matters.
In contrast, combination would be used if the order of the elements did not matter. For example, if the teacher only wanted to know the possible combinations of questions that could be included in a test, regardless of their order.
c. If a student doesn't show up to take the test, it does not affect the total number of outcomes. The number of outcomes is determined by the combination of questions and answers, not by the number of students present. Each student still has the possibility of different combinations of questions and answers, even if some of them don't participate.
d. If the teacher doesn't randomize the questions on the exam, it does not affect the total number of outcomes. The total number of outcomes is still determined by the possible combinations of questions and answers, regardless of their order. The lack of randomization may affect the fairness and accuracy of the exam, but it does not change the number of possible outcomes.
e. If each question on the test had the same answer (let's say C), the probability that students taking the test will get all the questions right would be 1. Since each question has the same answer, all the students would choose the correct answer for every question by default. The probability of getting all the questions right is certain or 100% in this case.