SOLUTION: 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 que

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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:
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)?
Answer by EdwinParker(16) About Me  (Show Source):
You can put this solution on YOUR website!
I will assume you are implying that the student will not leave 
any questions blank, and will therefore either gain or lose
some points for each of the N questions.

let:

N = the number of questions.
a = the number of points added for each correct answer.
b = the number of points subtracted for each incorrect answer.
p = the probability of answering any one question correctly.
C = the number of answers a student guesses correctly.
(Therefore the student guesses N-C answers incorrectly, since
I am assuming he or she leaves no answers blank.]
X = the student's score

Then the student's highest possible score is aN and the lowest
is -bN

The formula for the score X is

X = aC - b(N - C)

Solve that for C

X = aC - bN + bC

X + bN = aC + bC

X + bN = C(a + b)

%28X%2BbN%29%2F%28a%2Bb%29 = C

C = %28X%2BbN%29%2F%28a%2Bb%29

If that does not come out to a whole number, then the student
cannot possibly score exactly X, so you will then have to 
choose whether to round C up to the next higher integer so that
he or she makes slightly higher than C or to round C down to the
next lower integer and makes slightly lower than X.

The probability of scoring X (or as close to X as possible) 
is the probability of guessing C correct answers correctly,
which is:

%28matrix%282%2C1%2CN%2CC%29%29p%5EC%2A%281-p%29%5E%28N-C%29

where %28matrix%282%2C1%2CN%2CC%29%29 is the number of combinations
of N things taken C at a time.

Edwin