The problem states that a student randomly guessed the answers for two problems, each having four answer choices (a, b, c, d), and we are to find the probability that the student answered neither of the problems correctly.

Step-by-Step Solution:

1. Probability of answering a single question incorrectly:

   • Since there are 4 choices for each question and only 1 correct answer, the probability of getting the question wrong is: $P(\text{incorrect answer}) = \frac{3}{4}$

2. Probability of answering both questions incorrectly:

   • Since the events are independent (the outcome of one question does not affect the other), the probability of getting both answers wrong is: $P(\text{both incorrect}) = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16}$

Final Answer:

The probability that the student answered neither of the problems correctly is $\frac{9}{16}$.

Do you need further details or have any questions?

Here are five related questions to expand your understanding:

1. What would the probability be if the student guessed correctly on exactly one of the two problems?
2. How does the total number of questions influence the overall probability of incorrect answers?
3. How would the probability change if each question had 5 answer choices instead of 4?
4. What is the probability that the student answered at least one question correctly?
5. How would the probability change if the student answered 3 questions instead of 2?

Tip: When dealing with multiple independent events, you can multiply the probabilities of each event happening!