SOLUTION: A student is taking a standardized test consisting of multiple choice questions for which there are five options for each question. Seven points are awarded for each correct answer

Algebra ->  Probability-and-statistics -> SOLUTION: A student is taking a standardized test consisting of multiple choice questions for which there are five options for each question. Seven points are awarded for each correct answer      Log On


   

Question 311362: A student is taking a standardized test consisting of multiple choice questions for which there are five options for each question. Seven points are awarded for each correct answer, but the student loses 2 points for an incorrect answer. Questions left blank neither receive nor lose points. What is the minimum number of options that the student should be able to rule out before making a guess on any particular question?

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
A student is taking a standardized test consisting of multiple choice questions for which there are five options for each question. Seven points are awarded for each correct answer, but the student loses 2 points for an incorrect answer. Questions left blank neither receive nor lose points. What is the minimum number of options that the student should be able to rule out before making a guess on any particular question?

                                Points   Probability               
Rules out 0 and guesses right     7         1/5           
Rules out 0 and guesses wrong    -2         4/5

Expectation = 7(1/5)+(-2)(4/5) = -1/5

The expectation is negative, so one should not guess if only one
option can be eliminated.

                                Points   Probability           
Rules out 1 and guesses right     7         1/4
Rules out 0 and guesses wrong    -2         3/4

Expectation = 7(1/4)+(-2)(3/4) = 1/4       

Since the expected number is positive, the answer is 
"If 1 option can be eliminated, then guessing is the best policy.

If more options can be eliminated, then of course the best policy is to
guess. I'll go thorugh the others for instructive purposes, but it really 
isn't nexessary.  The minimum number of options that the student should 
be able to rule out before making a guess is 1. 

                                Points   Probability           
Rules out 2 and guesses right     7         1/3       
Rules out 0 and guesses wrong    -2         2/3

Expectation = 7(1/3)+(-2)(2/3) =  1

                                Points   Probability           
Rules out 3 and guesses right     7          1/2       
Rules out 0 and guesses wrong    -2          1/2

Expectation = 7(1/2)+(-2)(1/2) =  2.5 

                                 Points   Probability
Rules out 4 and guesses right     7           1       
Rules out 0 and guesses wrong    -2           0

Expectation = 7(1)+(-2)(0) =  7 

Edwin