Question

Suppose that you are taking a multiple choice test (consisting of only 4-answer-choice questions) for which you have mastered 60% of the material. When you actually take the test, if you know the answer of a question, then you will answer it correctly for sure; if you do not know the answer, however, you can still guess the answer (meaning that you will have 25% chance of getting the correct answer). Finally, assume that your answer to each question is independent of the others.

(a) For a given question on the test, what is the probability that you answer it correctly?

(b) Suppose that each multiple-choice question is worth of 10 points (assuming no partial credits). What is the probability that you will get at least 90 points (out of 100 points) over the 10 questions?

(c) Suppose that you enter a slightly different test. It is still of the same subject and the same format (i.e., 4-answer-choice questions). You will be given one question at a time. If you answered the question correctly, the test continues; if you answered a question wrong, the test stops. (Then you will be graded based on how many questions you can answer correctly.) What is the probability that you will have the chance to answer at least 3 questions?

(d) What is the average number of questions you will encounter in the test in (c)?

Answer 1

(a)

The probability that you answer the question correctly

= 0.6*1 + 0.4*0.25 = 0.7

(b)

To get atleast 90 points, atleast 9 questions should be correct

Thus, the required probability =

= 0.1493

(c)

To get chance to answer atleast 3 questions, the first answers need to be answered correctly

Thus, the required probability = = 0.49

(d) The average number of questions encountered

= 1 + 1/(1 - 0.7)

= 4.333