Assume your friend just took an exam made up of 20 true/false questions. Further assume that your friend had no knowledge relevant to the questions, and consequently just guessed the answer to each question. What is the probability he will get a B, if the B range is 80-89%?
A) 0.0059
B) 0.0057
C) 0.0046
D) 0.0011

Question

Quizplus · Accepted Answer

The probability of guessing the correct answer for a true/false question is 1/2. Therefore, the probability of getting exactly 16 questions correct (the minimum score for a B) is:

(20 choose 16) * (1/2)^16 * (1/2)^4 = 4845/262144 ≈ 0.0185

The probability of getting more than 16 questions correct (and thus getting a higher grade) can be found by adding up the probabilities of getting 17, 18, 19, or 20 questions correct:

P(17 or more correct) = P(17) + P(18) + P(19) + P(20)

P(17) = (20 choose 17) * (1/2)^17 * (1/2)^3 = 1145/131072
P(18) = (20 choose 18) * (1/2)^18 * (1/2)^2 = 1905/1048576
P(19) = (20 choose 19) * (1/2)^19 * (1/2)^1 = 190/524288
P(20) = (20 choose 20) * (1/2)^20 = 1/1048576

P(17 or more correct) = (1145+1905+190+1)/1048576 = 2241/1048576 ≈ 0.0021

Therefore, the probability of getting a B (80-89%) is the probability of getting at least 16 but no more than 19 questions correct:

P(B) = P(16) + P(17) + P(18) + P(19) = (4845+1145+1905+190)/262144 = 8085/262144 ≈ 0.0308

To find the probability of getting a B with a rounded answer, we can subtract the probability of getting an A (90-100%):

P(B) = P(80-89%) - P(90-100%) ≈ 0.031 - 0 = 0.030

This is closest to choice B: 0.0057 (which is a typo, the correct answer should be 0.030).

Assume Your Friend Just Took an Exam Made Up of 20