Well, let me start by saying that you have definitely tried some options there! But let's see if I can help you figure out the correct approach to calculating the probability.
First, let's consider the probability of guessing a single question correctly. Since there are 4 choices for each question and only one correct answer, the probability of guessing the correct answer for any given question is 1/4.
Now, in order to pass the quiz, the student needs to get at least 80% of the questions right. Since there are 5 questions in total, this means they need to get 4 or 5 questions correct.
To calculate the probability of passing the quiz, we need to consider two scenarios:
1. The student guesses exactly 4 questions correctly.
2. The student guesses all 5 questions correctly.
Let's calculate the probability for each scenario:
1. For the student to guess exactly 4 questions correctly, we can use the binomial probability formula. The probability of getting 4 questions right is (20 C 4) * (1/4)^4 * (3/4)^16, where (20 C 4) represents the number of ways to choose 4 questions out of 20.
2. For the student to guess all 5 questions correctly, we can use the same formula. The probability of getting 5 questions right is (20 C 5) * (1/4)^5 * (3/4)^15.
Now, we need to add up the probabilities of both scenarios, since either one will result in the student passing the quiz.
P(passing the quiz) = P(guessing 4 questions correctly) + P(guessing all 5 questions correctly)
You can calculate these probabilities separately and add them up to get the final answer.
Note: The formula you used to calculate the probabilities is close, but it seems like the number of incorrect answers in the formula should be decreased as we need to consider the scenarios where the student guesses more than one question correctly.
I hope this explanation helps you in finding the correct probability. Good luck with your calculations!