Science:Math Exam Resources/Courses/MATH103/April 2011/Question 02 (d)

MATH103 April 2011

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[hide] 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.) What is the probability that a student who did not study and randomly checks his/her answers still passes the test? With what probability does the student have to get every answer correct in order to get a perfect score with a probability of at least 80%?

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.

[show] Hint 1
This is a sequence of 6 Bernoulli trials.

[show] 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.

[show] 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 $p$ 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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

[show] Solution 1

This is a Bernoulli trial, where each multiple-choice 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 $p=1/4$ and his probability of failure is $q=3/4.$ Passing the test means that he obtained either 5 or 6 successes in 6 trials. Let $X$ be his total score. The probability that he passes is ${\begin{aligned}P({\text{pass}})&=P(X=5\ or\ X=6)\\&=P(X=5)+P(X=6)\\&=C(6,5)p^{5}q+C(6,6)p^{6}\\&=6(1/4)^{5}(3/4)+(1/4)^{6}\\&={\frac {6\cdot 3+1}{4^{6}}}\\&={\frac {19}{4^{6}}}\end{aligned}}$ Note: here $C(n,\,k)$ is the binomial coefficient, $C(n,k)=C_{k}^{\,n}={\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.$ (This is the number of ways to choose k objects among n total objects.)

[show] Solution 2
The probability of getting a perfect score is $p^{6}$, where $p$ is the probability of success on a single question. In order for $p^{6}$ to be at least 80%=4/5, $p$ must be at least $(4/5)^{1/6}.$ (This is approximately 96%, but it would be fine to leave your answer in the form $(4/5)^{1/6}$ on the exam.)

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