If the test was made without any regard to distributions/patterns, then probability could not help you because each question would be independent and uniformly distributed. This means that nothing can be inferred about one answer based on information about the other answers, and that no answer is more likely than the others. However, I would argue that this is a less likely scenario becuase humans are notoriously bad random number generators. Making the answers be truly i.i.d. would usually require the use of a PRNG or similar means. This seems much more likely in the case of standardized tests, and somewhat less likely when you receive simple "circle the correct answer"-style tests (as opposed to OMR-style answer sheets)
So let's assume that the answers are either not independent, or that they are not uniformly distributed. Now can you use probability to help you? Let's look at each case:
The professor may favor some answers more than others
If the distribution of answers is not uniform, then for any given problem there are answers that are more likely to be correct than others. Notice that this has nothing to do with patterns, or independence. You could easily generate a test where the answers are independent but not uniformly distributed. For example, roll a six-sided dice, and assign the answers as follows:
1=A 2=B 3=D 4=C 5=C 6=C. You will notice that
C is the most likely answer for any given question but the dice rolls are still completely independent.
Can you use this information to help you take the test? Maybe. If you knew before ever seeing the test that this particular professor favors
C as an answer, and that previous tests from this professor had
C as the correct answer more than 50% of the time, you could potentially leverage this information. But how? If I know that any given question has a probability of 0.5 of having
C as the correct answer, but I observe that in my completed test only 40% of the answers are
C, the best I can do is to re-evaluate all of the non-
C answers, and see if I think that
C might be a better answer. This will not be very fruitful, especially because the fact
C is more likely does not mean that if you have more
Cs that you have done anything wrong.
C can be more likely to occur than
B and yet
B can show up more frequently on a given test.
Instead, the best way to use this information is on a per-problem basis. Read the problem and the answer choices. Throw out any answers that are obviously not correct, and then evaluate the ones that might be correct. From the remaining answers you can combine the known distribution with your knowledge about the problem to create a likelihood for each answer. The answer with the highest likelihood wins.
The professor may avoid long sequences of the same answer
In this case, you can use a conditional probability that a
The professor may avoid seemingly "non-random" patterns
B answer will occur given that the previous
n answers were
B. You can again combine this information with your knowledge about the problem, and then maximize the likelihood. The reason that this won't work is that you would need very reliable information about how likely the professor is to allow 2
Bs in a row, 3
Bs, etc. This information could possibly be extracted if you had a large enough sample size of prior tests from this same professor, but without getting inside the professors brain, it would still require a lot of inference on your part. It is entirely possible that this professor has no problem with allowing 6
Bs in a row, but since such a sequence only has a 0.02% chance of occurring naturally, you may collect 1000 prior tests and still never see this pattern.
The arguments against this one are at least as strong as the previous. Without specific knowledge of what the professor considers to be a "pattern", and to what degree he is willing to allow patterns to occur naturally, it would be nearly impossible to extract this information from prior tests.
The professor may attempt to create a "perfect" uniform distribution
In the extreme case, you might know that there were exactly the same number of each answer on the test. If you ended up with two more
Cs you could assume that one of the
Bs was meant to be a
C, and attempt to locate the offender.
Having said that, this won't work with anything short of a "perfect" distribution in which each answer is represented exactly the same number of times. If the professor just has a soft goal of trying to distribute the answers evenly, then you really can infer almost nothing.
There is very little to be gained from "probabilistic test taking" unless you have access to information about how the test is generated. Unless the professor has explicitly stated the rules he uses to distribute the answers, the only way to obtain this information is through statistical inference after processing "sufficient" historical data. As described above, this may not be just impractical, but impossible. The professor may not have written enough tests in his lifetime to make reasonable inferences about how his tests are generated.
But hey, when in doubt, mark
C. That's what I do.