Researchers from the Educational Testing Service (ETS) found that providing immediate feedback to students answering open ended questions can dramatically improve students' future performance on exams (Educational and Psychological Measurement, Feb. 2010). The ETS researchers used questions from the Graduate Record Examination (GRE) in the experiment. After obtaining feedback, students could revise their answers. Consider one of these questions. Initially, 50% of the students answered the question correctly. After providing immediate feedback to students who answered incorrectly, 70% answered correctly. Consider a bank of 100 open-ended questions similar to those on the GRE.
a. In a random sample of 20 students, what is the probability that more than half initially answer the question correctly?
b. Refer to part a. After providing immediate feedback, what is the probability that more than half of the students answer the question correctly?