True/False and Multiple Choice TestsDate: 05/24/99 at 22:00:14 From: huxley Subject: Permutations/combinations Here are my questions: 1) There is a test with 20 true/false questions. If the passing grade is 14, what's the probability that a student will pass the test by guessing the answers? Is .5^14 correct? 2) Another teacher says she can reduce the number of questions but give the students more choices per question. Is the teacher correct? If so, how many questions and how many choices per question should the teacher give to get roughly the same probability of passing by guessing? What should the passing grade be? If the teacher is wrong, explain why it is not possible to get the same probability of passing with fewer questions. Date: 05/25/99 at 10:09:51 From: Doctor Anthony Subject: Re: Permutations/combinations >1) There is a test with 20 true/false questions.... This is binomial probability with n = 20, p = 1/2, q = 1/2 To pass you require probability P(14)+P(15)+ ....+P(20) P(14) = C(20,14) x (1/2)^14 x (1/2)^6 = C(20,14) x (1/2)^20 Clearly the total probability is (1/2)^20 x [C(20,14)+C(20,15)+C(20,16)+.....+C(20,20)] = (1/2)^20 x 60460 = 0.057659 >2) Another teacher says she can reduce the number of questions... There are hundreds of possibilities. I will deal with one. Suppose we have 10 questions with 5 choices of answer per question. Now there is 1/5 probability of guessing correctly and 4/5 probability of guessing incorrectly. Probability of 3 correct = C(10,3) x (1/5)^3 x (4/5)^7 = 0.201326 4 correct = C(10,4) x (1/5)^4 x (4/5)^6 = 0.088080 5 correct = C(10,5) x (1/5)^5 x (4/5)^5 = 0.026424 6 correct = C(10,6) x (1/5)^6 x (4/5)^4 = 0.005505 7 correct = C(10,7) x (1/5)^7 x (4/5)^3 = 0.000786 8 correct = C(10,8) x (1/5)^8 x (4/5)^2 = 0.0000737 9 correct = C(10,9) x (1/5)^9 x (4/5) = 0.0000041 10 correct = (1/5)^10 = 0.0000001 From these results we can see that 5 or more correct has probability of approximately 0.03, while 4 or more correct has probability of roughly 0.1. With 20 questions and 14 or more correct the probability was approximately 0.06, so in the second situation we have devised a test with less probability of passing if 5 or more correct answers are required but greater probability of passing if 4 or more correct answers are required. You could change the number of questions to, say, 12 or 15 and the number of choices of answer per question to 3 or 4 to get another set of probabilities. The possible variations are immense. For example, in the extreme you could have 1 question with 17 possible answers giving you a probability of 0.0588 of passing - nearly the same probability as the 20-question test with a requirement of 14 or more correct answers. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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