Question

A standardized test consists of 100 multiple-choice questions. Each question has five possible answers, only one of which is correct. Four points are awarded for each correct answer. To discourage guessing, one point is taken away for every answer that is not correct (this includes answers that are missing).

The company that creates the test has to understand how well a student could do just by random guessing. Suppose a student answers each question by picking one of the five choices at random independently of the choices on all other questions. Let S be the student's score on the test.

a) Find ?(S).

b) Find P(S>10). Write your answer as a math expression, then use the code cell below to find its numerical value and provide it along with your math expression.

Answer 1

a)

Let the random variable X = marks obtained in a question

S	P(S)
4	1/5=0.2
-1	4/5=0.8

The expected value of marks is obtained using the formula,

Using the property of expectation, the expected value for 100 questions is,

b)

To get the exam score, S = 10, 22 questions should be correct

Exam score S = 22x4+(-1)x78=10

Let the random variable Y = number of the correct answers in the exam

The sample size of 100 questions will have a binomial distribution with parameter n and p. The distribution can be represented as;

Now, the required probability,

The probability is obtained in excel using the function =BINOM.DIST(22,100,0.2,TRUE)

Hence the probability of getting more than 10 marks is 0.2611