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Ann plans to just guess answers on a test she knows nothing about. If the test has three questions with three multiple choice answers per question, what is the probability she will guess two of three questions correctly or three out of three questions correctly?

Correct Answer:

**D**Multiply the number of arrangements by the arrangement probability

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.