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**D**

One approach to solving this problem is as follows:

- Write the general probability equation for the problem
- Determine the probability for selecting two correct answers and one incorrect answer for one arrangement

Determine the number of arrangements that one can obtain two correct and one incorrect answer - Multiply the number arrangements by the probability of one arrangement
- Repeat the procedure for obtaining three out of three correctly
- Substitute the values obtained in steps 3 and 4 into the general probability equation of step 1

- Write the general probability equation for the problem

P(2 of 3 OR 3 of 3 Correct) = P(2 of 3 Correct) + P(3 of 3 Correct) - Determine the individual probability for selecting two correct answers and one incorrect answer

The probability of guessing an answer correctly for one question

P(Correct) = 1/3

The probability of not guessing an answer correctly for one question is

P(Not Correct) = 2/3

The probability of getting one combination where 2/3 of the answers are correct and 1/3 of the answers are incorrect is:

P(2 of 3 Correct for One Arrangement) = P(Correct)*P(Correct)*P(Not Correct) =(1/3)(1/3)(2/3) = 2/27 - Determine the number of arrangements that one can obtain two correct answers and one incorrect answer

Because there is more than one way to get 2 problems correct and 1 problem incorrect, the number of different arrangements must be determined.

This can be done through the construction of a probability table, below, which shows the different ways one can score on the test. Since there are 3 questions, and two ways to answer a question (incorrectly or correctly), there are 2^{3}or 8 ways to answer the questions incorrectly or correctly. There will be eight entries in the table.

Table: Different Combinations of Correct and Incorrect Answers for a test with 3 questions and 3 multiple choice answers, 0 is incorrect, 1 is correct

Answer 1 Answer 2 Answer 3 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1

P(2 of 3 Correct) = 3*P(One Combination of 2 of 3 Correct) = 3* P(Correct)*P(Correct)*P(Not Correct) =(1/3)(1/3)(2/3) = 6/27 = 2/9. - Repeat the procedure for selecting three out of three correctly

There is only one possibility where one can correctly guess all three answers.

P(3 of 3 Correct) = 1* P(One Combination of 3 of 3 Correct) = 1*P(Correct)*P(Correct)*P(Correct) = (1/3)(1/3)(1/3) = 1/27 - Substitute the values obtained in steps 3 and 4 into the general probability equation of step 1

P(2 of 3 Correct) OR P(3 of 3 Correct) = P(2 of 3 Correct) + P(3 of 3 Correct) = 6/27 + 1/27 = 7/27

So the correct answer is 7/27, Answer D.

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