Question

In a multiple-choice test, each question has four options. Students will get 2 points for each correct answer; lose 1 point for each incorrect answer; and receive no points for unanswered questions. A student does not know the correct answer for one question. Is it to her advantage or disadvantage to guess an answer? Show your calculations for expected value. Show work.

Answer 1

expected value right?

Answer 2

yes

Answer 3

suppose you guess. then the probability you get it right is \(\frac{1}{4}\), and the probability you get it wrong is \(\frac{3}{4}\)

Answer 4

before we compute, it is pretty obvious that it is a bad idea to guess since you only get it right one time out of 4, and will lose 1 point 3 times out of 4, but we can compute the expected value
it is what you get, times the probability you get it added up
so we compute
\[2\times \frac{1}{4}-1\times \frac{3}{4}=\frac{2-3}{4}=-\frac{2}{4}=-\frac{1}{2}\]

Answer 5

ok that was wrong!!

Answer 6

\[\frac{2}{4}-\frac{3}{4}=-\frac{1}{4}\]

Answer 7

in any case it is negative, so you do not want to guess

Answer 8

easier way is to imagine you answer all questions
you get one right for two points , three wrong for -3 points, and your total is then -1 point
averaged over the 4 answers you get \(-\frac{1}{4}\) per answer

Answer 9

in other words, expected value is really a kind of average