1. A multiple choice test has 10 questions. Each question has five possible answers and only one of them is correct. One very bad student refuses to study for this test. Instead, for each question, he chooses one of the selections randomly.

a) How many different possible ways can this student fill out the test?

Each question has 5 possible answers, and so there are 5*5*...*5 = 5 raised to the 10th power ways of answering.
 
 

b) The student needs to get 6 or more correct in order to pass. What's the probability he will pass? (Don't give an actual number; just give the equation.) What assumptions are you making?

Let X represent the number the student gets correct in 10 attempts.  We want P(X >= 6) = P(X = 6) + P(X = 7) + ... + P(X = 10)
where P(X = x) = (10 choose x) (1/5)^x  (4/5)^(10-x)
where x^y means x raised to the yth power.

For this probability calculation to be correct, we must assume:
a) only two possible outcomes for each attempt (right & wrong)
b) the probability of getting one right is the same for each question (so for each question the correct answer must be equally likely to be any of the 5 choices.)
c) The attempts must be independent (this could be spoiled if there were a pattern that the student might guess, like "c" was the correct answer too often.)
d) There are  a fixed number of trials.  (There are precisely 10 questions on the test.)