Effectiveness of online and face-to-face learning. A study published in the e-Journal of Business Education & Scholarship of Teaching (Vol. 12, 2018) compares the effectiveness of online learning versus face-to-face learning. The researchers observed a Principles of Macroeconomics class taught twice by the same professor to two different student groups. One group was taught face to face and the other group was taught online. Both groups were tested for their knowledge of the subject before the class (pretest) and after the class (post-test) to determine what they had learned, and the test scores were recorded. Consider the following scenarios where five students are randomly selected from the group that was taught online and you are the last of the five students to take the test. Your group passes the test if the majority of the group pass the test.

a. Determine the number of possible outcomes of each of the five students’ post-test results. Each student can either pass or fail. (These outcomes represent the sample points for the experiment.)

b. Assume each student is equally likely to pass or fail. Assign reasonable probabilities to the sample points in part

a. c. Your test result is the decisive result for the group if out of the remaining four students, two have passed and two have failed. Assuming you are student number five, how many of the sample points in part

a. result in a 2–2 split for the other four students?

d. Use your answers from parts a–c to find the probability that your result is the decisive result for the group.

e. Now, to make your result count more, you form a subgroup with two other students and ask to be marked “jointly” – if the majority (i.e., 2 or 3 of your sub-group) pass/fail then all of you pass/fail. This means the result of the sub-group will always determine the result of the group of five. Find the probability that your test result is the decisive result for the group