An exam is comprised of true or false questions Students ask

An exam is comprised of true or false questions. Students ask any question that they know the answer to. For the other questions they guess the answer. What is the probability that a student knew the answer if he answered it correctly?
I\'m under the impression I have to write Baye\'s formula for this, but I\'m struggling to show my work. Any explanation for what each part of the formula represents in the context of this problem would be greatly appreciated.

Solution

Lets suppose there are 10 items on a true-false quiz.

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