If you make random guesses for 10 multiple-choice test questions (each with five possible answers), what is the probability of getting at least 1 correct? if a very lenient instructor says that passing the test occurs if there is at least one correct answer, can you reasonably expect to pass by guessing?

There is a 1/5 chance of getting each question correct.  The probability of getting at least 1 correct is written as such:
P(X ≥ 1) = 1 - P(X = 0) 
This is because the only other option than getting at least one correct is to get nothing correct.
Since there is a 1/5 chance of getting each question correct, there is a 4/5 chance of getting each question wrong:
1 - P(X = 0) = 1 - (4/5)^10 (since there are 10 questions, with a 4/5 chance of missing each one)
= 1 - 0.1073741824 ≈ 0.8926 = 89.26% chance of getting at least one question correct.  Given this probability, it is reasonable to expect to pass by guessing.