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True/False and Multiple Choice Tests

Date: 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 

- Doctor Anthony, The Math Forum   
Associated Topics:
High School Probability

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