A multiple-choice quiz has 20 questions. Each question has five possible answers

Question

A multiple-choice quiz has 20 questions. Each question has five possible answers, of which only one is correct.

A) What is the probability that sheer guesswork will yield at least 10 correct answers?

B) What 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

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 20 * 0.2
= 4
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 20 * 0.2 * 0.8
= 3.2
III.
standard deviation = sqrt( variance ) = sqrt(3.2
=1.78885
PART A.
P( X < 10) = P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0) +
= ( 20 9 ) * 0.2^9 * ( 1- 0.2 ) ^11 + ( 20 8 ) * 0.2^8 * ( 1- 0.2 ) ^12 + ( 20 7 ) * 0.2^7 * ( 1- 0.2 ) ^13 + ( 20 6 ) * 0.2^6 * ( 1- 0.2 ) ^14 + ( 20 5 ) * 0.2^5 * ( 1- 0.2 ) ^15 + ( 20 4 ) * 0.2^4 * ( 1- 0.2 ) ^16 + ( 20 3 ) * 0.2^3 * ( 1- 0.2 ) ^17 + ( 20 2 ) * 0.2^2 * ( 1- 0.2 ) ^18 + ( 20 1 ) * 0.2^1 * ( 1- 0.2 ) ^19 + ( 20 0 ) * 0.2^0 * ( 1- 0.2 ) ^20 +
= 0.99741
P( X > = 10 ) = 1 - P( X < 10) = 0.00259

PART B.
mean = 20 * 0.2 = 4

PART C.
If we award 5 points on correct question, that is for expected mean of 4, the calculated
marks be 5*4 = 20 & the remaing unanswered left are = 16 and to maintain a zero he has to
wrongly ansswered 16 questions, by this the negative amuont calculation be = 20/16 = 1.25 to
maintain a zero

PART D.
expected mean = mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 20 * 0.25
= 5

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