Few students manage to complete their schooling without taking a standardized ad
ID: 3065922 • Letter: F
Question
Few students manage to complete their schooling without taking a standardized admissions test such as the Scholastic Achievement Test, or SAT (used for admission to college); the Law School Admissions Test, or LSAT; and the Graduate Record Exam, or GRE (used for admission to graduate school). Sometimes, these multiple-choice tests discourage guessing by subtracting points for wrong answers. In particular, a correct answer will be worth +1 point, and an incorrect answer on a question with five listed answers (a through e) will be worth 1 4 point.
(a) Find the expected value of a random guess.
(b) Find the expected value of eliminating one answer and guessing among the remaining four possible answers. (Enter an exact number as an integer, fraction, or decimal.)
(c) Find the expected value of eliminating three answers and guessing between the remaining two possible answers. (Enter an exact number as an integer, fraction, or decimal.)
(d) Use decision theory and your answers to parts (a), (b), and (c) to create a guessing strategy for standardized tests such as the SAT.
a) Random guessing is a winning strategy.
b) There is no winning strategy for guessing.
c) If you can eliminate no less than three answers, guessing is a winning strategy.
d) If you can eliminate one or more answers, guessing is a winning strategy.
Explanation / Answer
a) The expected value of guessing a question would be computed as:
= 1*Probability of getting the question correct - 0.25*Probability of getting the question incorrect
As there are 5 options in the question, probability of getting a question correct would be 0.2 and incorrect would be 0.8. Therefore we get the required probability as:
= 1*0.2 - 0.25*0.8
= 0.2 - 0.2 = 0
Therefore 0 is the expected value here.
b) Now as we have removed one of the option, probability of getting it correct here would be 1/4 = 0.25 and therefore 1- 0.25 = 0.75 would be the probability of getting it incorrect. Therefore the expected value of attempting a question here would be computed as:
= 1*0.25 - 0.25*0.75
= 0.25 - 0.1875
= 0.0625
Therefore 0.0625 is the required expected value here
c) Now here we have removed 3 options, therefore the probability of getting it correct would be 1/2 = 0.5 and probability of getting it incorrect would also be 0.5. Therefore the expected value of attempting a question here would be computed as:
= 1*0.5 - 0.25*0.5
= 0.5 - 0.125
= 0.375
Therefore 0.375 is the required expected value here
d) From the above computations we see that we always get an expected value if we are able to eliminate at least one of the 5 options. Therefore If you can eliminate one or more answers, guessing is a winning strategy is the correct strategy here.
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