6) A multiple-choice quiz has 20 questions. Each question has five possible answ

Question

6) A multiple-choice quiz has 20 questions. Each question has five possible answers, of which only one is correct. A) W hat is the probability that sheer guesswork will yield at least 10 correct answers? B) W hat is the expected number of correct answers by sheer guesswork? C) Suppose 5 points are awarded for a correctly answered question. How many points should be deducted for an incorrectly answered question, so that for a student guessing randomly, the expected score on a question is zero? (Most standardized tests use this method to set penalties for guessing). D) If a student is able to correctly eliminate one option as a possible correct answer but is still guessing randomly, what happens to his/her expected score for that question? Use your answer to C) as the number of points being deducted for an incorrect answer E) If a student is able to correctly eliminate two options as possible correct answers but is still guessing randomly, what happens to his/her expected score for that question? Use your answer to C) as the number of points being deducted for an incorrect answer.

Explanation / Answer

Solution:-

p = 1/5 = 0.20

n = 20

a) The probability that sheer gueswork will yield at least 10 correct answers is 0.0026.

p = 1/5 = 0.20

n = 20

By applying binomial distributiion:-
P(x,n) = nCx*px*(1-p)(n-x)

P(x > 10) = 0.0026

b) The expected number of correct answers by sheer guesswork is 4.

E(x) = n × p

E(x) = 20 × 0.20

E(x) = 4

c) The points deducted for the incorrect answers is 1.25.

E(x) = 0(Given)

E(x) = p × 5 + (1 - p) × S

0 = 0.20 × 5 + 0.80 × S

S = - 1.25

d) Expected score for this question is 0.3125.

p = 1/4 = 0.25

1 - p = 0.75

E(x) = p × 5 + (1 - p) × S

E(x) = 0.25 × 5 - 0.75 × 1.25

E(x) = 1.25 - 0.9375

E(x) = 0.3125

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