Question 209316: Suppose there are 10 items on a true-false quiz. The person taking the test does not read the questions; he just answers the questions randomly. What is the probability of his guessing all answers correctly?
How should I find out how many answers he answered correctly??
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! true false quiz has 2 possible answers for each question meaning that the probability of getting the correct answer is .5 and the probability of getting the wrong answer is .5.
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the probabiliy of getting 1 correct answeer is .5
getting 2 is .5 * .5 = .5^2 = .25
getting 3 is .5 * .5 * .5 = 5^3 = .125
probability of getting all 10 correct is .5^10 = .000976563
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you would not know exactly how many he answered correctly, but the odds are that he probably answered 50% correctly.
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let' see what the distribution is with 3 questions.
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odds are (.5*.5*.5) to get all 3
odds are (.5*.5*.5) to get none.
odds are (.5*.5*.5) * 3!/1!2! to get exactly 1 correct.
odds are (.5*.5*.5) * 3!/2!1! to get exactly 2 correct.
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if the total probability is 1, then we did this correctly.
total probability is:
all 3 = .125
none = .125
1 = .125 * 3 = .375
2 = .125 * 3 = .375
.125 + .125 + .375 + .375 = 1
this means we did this correctly.
if you were a betting man, you would bet that out of 3 questions, he would answer 1 or 2 correctly because those have the highest probability of occurring giving you a 75% chance of being right.
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extending this to 10 questions, the odds that he would answer 1 or 2 or 3 or 4 or 5, etc correctly are given as follows:
0 correct = .5^10 * 1 = .0009765625
1 correct = .5^10 * 10!/9!1! = .0009765625 * 10 = .009765625
2 correct = .5^10 * 10!/8!2! = .0009765625 * 45 = .043945313
3 correct = .5^10 * 10!/7!3! = .0009765625 * 120 = .1171875
4 correct = .5^10 * 10!/6!4! = .0009765625 * 210 = .205078125
5 correct = .5^10 * 10!/5!5! = .0009765625 * 252 = .24609375
6 correct = .5^10 * 10!/6!4! = .0009765625 * 210 = .205078125
7 correct = .5^10 * 10!/7!3! = .0009765625 * 120 = .1171875
8 correct = .5^10 * 10!/8!2! = .0009765625 * 45 = .043945313
9 correct = .5^10 * 10!/9!1! = .0009765625 * 10 = .009765625
10 correct = .5^10 * 1 = .0009765625
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if we did this correctly, then the total probability will be equal to 1.
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the sum of all these probabilities is equal to 1 which means we probably did it correctly.
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if you were a betting man you would bet that he got 5 right and you would have about a 25% change of winning.
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if you bet that he got 4 to 6 right, then you would have about a 65% chance of winning.
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if you bet that he got 3 to 7 right, then you would have about a 87% chance of winning.
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only other way you could figure what his score was would be if he answered all questions true or he answered all questions false. then you simply have to look at the answers to the test and you can figure out how he scored.
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if he did that (answer all true and answer all false), and he didn't tell you which, then you would have a 50% probability of guessing which one he used and what his score was based on that.
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alternately you could look at both options and you would know that he got either x right or y right depending on whether he picked true for all the questions or he picked false for all the questions.
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of course, you would need to know the answers to the questions in order to figure this out.
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Answer by MathTherapy(10555)  (Show Source):
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