document.write( "Question 616887: Hello, I'm having some problems related to test exams and probability. Say you have a test exam composed of N multiple answer questions (I have the probability of guessing each question right or wrong by chance pretty much covered, that's easy). Each question can be either correct (which will add a certain amount of points to the final score), incorrect (which will substract another quantity from the final score) or blank (which will not add nor substract). My question is:
\n" ); document.write( "If we know the probability of guessing each question, what is the probability of achieving a certain final score guessing all the test (eg. 5.75)? \n" ); document.write( "

Algebra.Com's Answer #388019 by EdwinParker(16) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "I will assume you are implying that the student will not leave \r\n" );
document.write( "any questions blank, and will therefore either gain or lose\r\n" );
document.write( "some points for each of the N questions.\r\n" );
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document.write( "let:\r\n" );
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document.write( "N = the number of questions.\r\n" );
document.write( "a = the number of points added for each correct answer.\r\n" );
document.write( "b = the number of points subtracted for each incorrect answer.\r\n" );
document.write( "p = the probability of answering any one question correctly.\r\n" );
document.write( "C = the number of answers a student guesses correctly.\r\n" );
document.write( "(Therefore the student guesses N-C answers incorrectly, since\r\n" );
document.write( "I am assuming he or she leaves no answers blank.]\r\n" );
document.write( "X = the student's score\r\n" );
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document.write( "Then the student's highest possible score is aN and the lowest\r\n" );
document.write( "is -bN\r\n" );
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document.write( "The formula for the score X is\r\n" );
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document.write( "X = aC - b(N - C)\r\n" );
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document.write( "Solve that for C\r\n" );
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document.write( "X = aC - bN + bC\r\n" );
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document.write( "X + bN = aC + bC\r\n" );
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document.write( "X + bN = C(a + b)\r\n" );
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document.write( " = C\r\n" );
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document.write( "C = \r\n" );
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document.write( "If that does not come out to a whole number, then the student\r\n" );
document.write( "cannot possibly score exactly X, so you will then have to \r\n" );
document.write( "choose whether to round C up to the next higher integer so that\r\n" );
document.write( "he or she makes slightly higher than C or to round C down to the\r\n" );
document.write( "next lower integer and makes slightly lower than X.\r\n" );
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document.write( "The probability of scoring X (or as close to X as possible) \r\n" );
document.write( "is the probability of guessing C correct answers correctly,\r\n" );
document.write( "which is:\r\n" );
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document.write( "where  is the number of combinations\r\n" );
document.write( "of N things taken C at a time.\r\n" );
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document.write( "Edwin

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