Question

In: Statistics and Probability

1. On a multiple choice test where each question has 4 choices of which only one...

1. On a multiple choice test where each question has 4 choices of which only one is correct you get 1.8 points for a correct answer and you lose .75 points for an incorrect answer. You lose no points nor gain any points if you leave the problem blank. We let the random variable x = points earned from guessing on a question.

a) Find the probability distribution for x

b) Is guessing a good idea if you cannot eliminate any responses? Show work

c) If you can eliminate one choice is guessing a good idea? Show work

Solutions

Expert Solution

a)

Since, we are guessing on a question, we are attempting the problem and are not leaving it blank. Now there are two possible values of x, namely x = 1.8 if we guess the answer correctly and x = -0.75 (minus sign represents that we lost marks) if we guess the wrong answer.

Now, P(x=1.8) = P(guessing the correct answer) = (No. of correct options) / (Total no. of options) = 1/4

P(x=-0.75) = P(guessing the incorrect answer) = 1 - P(guessing the correct answer) = 1 - 1/4 = 3/4

Thus, the probability distribution of x is given by:
P(x=1.8) = 1/4 ; P(=-0.75) = 3/4 [ANSWER]

b)

If we cannot eliminate any response, then we have 4 choices out of which we choose one option.

Thus,

P(x=1.8) = P(guessing the correct answer) = (No. of correct options) / (Total no. of options) = 1/4

P(x=-0.75) = P(guessing the incorrect answer) = 1 - P(guessing the correct answer) = 1 - 1/4 = 3/4

Thus, the expected value of x is given by:

Since, the expected value of x comes out to be negative, guessing is not a good idea since on an average we expect to score to be around the expected value of x and since this value is negative, we expect (on an average) to lose marks if we guess.

c)

If we can eliminate one choice, then we have 3 choices out of which we choose one option.

Thus,

P(x=1.8) = P(guessing the correct answer) = (No. of correct options) / (Total no. of options) = 1/3

P(x=-0.75) = P(guessing the incorrect answer) = 1 - P(guessing the correct answer) = 1 - 1/3 = 2/3

Thus, the expected value of x is given by:

Since, the expected value of x comes out to be positive, guessing is a good idea since on an average we expect to score to be around the expected value of x and since this value is positive we expect (on an average) to gain marks if we guess.

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