Science:Math Exam Resources/Courses/MATH103/April 2011/Question 02 (d)
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q5 (c) • Q5 (d) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q7 (c) • Q8 •
Question 02 (d) 

A student takes a multiple choice test with 6 questions each of which has 4 possible answers and exactly one is correct. To pass the test at least 5 correct answers are required (Note: Simplify your answers as much as possible but leave fractions and powers.)

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

This is a sequence of 6 Bernoulli trials. 
Hint 2 

In part (1), what is the probability of success (i.e. answering correctly) on each test question? If he passes the test, then he must have obtained either 5 or 6 successes in total. 
Hint 3 

Part (2) could be interpreted as follows: What must be the student's probability of success on each question, if he is to have an 80% probability of getting a perfect score? (That is, what must the probability of success in a single Bernoulli trial be, if the probability of 6 successes in 6 trials is at least 80%?) 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. This is a Bernoulli trial, where each multiplechoice question is a trial. For each trial, "success" means that the student answered the question correctly and "failure" means that he answered the question incorrectly. If the student randomly checks answers (with the same probability for each of the 4 responses), then his probability of success on each question is and his probability of failure is Passing the test means that he obtained either 5 or 6 successes in 6 trials. Let be his total score. The probability that he passes is Note: here is the binomial coefficient, (This is the number of ways to choose k objects among n total objects.) 
Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. The probability of getting a perfect score is , where is the probability of success on a single question. In order for to be at least 80%=4/5, must be at least (This is approximately 96%, but it would be fine to leave your answer in the form on the exam.) 