A university course requires students to complete four multichoice tests. Each test has 10 questions with 5 possible answers to each question. In this question we’ll investigate whether a student can pass the course just by guessing.

a. Consider just one of the four tests. Suppose a student guesses each answer completely at random. Let X be the number of correct answers. What is the distribution of X?

b. Plot the distribution of X using a barplot in R, and specify the command you used.

c. The pass mark for a single test is 5 out of 10. What is the probability the student passes? Write it as a probability statement, give the R command for the calculation, and find the answer.

d. There are four tests in total, making a total of 40 questions.

The student guesses in the same way for all four tests.

Let T be the student’s total mark out of 40. What is the distribution of T?

e. What is the probability that the student attains the pass mark of 20 out of 40, by pure guesswork? Write it as a probability statement, give the R command, and find the answer. Interpret your result in terms of whether it’s a good idea for a student to use the guessing strategy.

f. Now suppose a student knows enough to be able to exclude two wrong answers from each question. They now only have to guess among the other three answers. Define a suitable random variable, specify its distribution, and find the chance of passing the course in this new scenario. Do you recommend this strategy?