A multiple-choice test has 10 questions. Each question has 5 choices. If a student answers each question by guessing randomly, find the probability that the student gets:

5,7

To find the probability that the student gets a certain number of questions correct, we need to use the concept of probability in combinatorics. Specifically, we can use the binomial probability formula. Given that each question has 5 choices, the probability of getting a particular question correct by random guessing is 1/5, while the probability of getting it wrong is 4/5.

a) To find the probability that the student gets at most 4 questions correct, we need to calculate the probability of getting 0, 1, 2, 3, or 4 questions correct and then sum them up.

- Probability of getting 0 correct answers:
For each question, the probability of getting it wrong is 4/5.
P(0 correct) = (4/5)^10

- Probability of getting 1 correct answer:
To calculate the probability of getting exactly 1 correct answer, we multiply the probability of getting 1 answer correct (1/5) by the probability of getting the remaining 9 answers wrong (4/5)^9.
P(1 correct) = C(10,1) * (1/5) * (4/5)^9

Likewise, we calculate the probabilities for getting 2, 3, and 4 correct answers using the formula:
P(k correct) = C(10,k) * (1/5)^k * (4/5)^(10-k)

Finally, we add up all the individual probabilities to find the probability of getting at most 4 questions correct:
P(at most 4 correct) = P(0 correct) + P(1 correct) + P(2 correct) + P(3 correct) + P(4 correct)

b) To find the probability of getting at least 7 questions correct, we need to calculate the probabilities of getting 7, 8, 9, or 10 questions correct, and then sum them up.

- Probability of getting 7 correct answers:
P(7 correct) = C(10,7) * (1/5)^7 * (4/5)^3

- Probability of getting 8 correct answers:
P(8 correct) = C(10,8) * (1/5)^8 * (4/5)^2

- Probability of getting 9 correct answers:
P(9 correct) = C(10,9) * (1/5)^9 * (4/5)^1

- Probability of getting 10 correct answers:
P(10 correct) = C(10,10) * (1/5)^10 * (4/5)^0

Finally, we add up all the individual probabilities to find the probability of getting at least 7 questions correct:
P(at least 7 correct) = P(7 correct) + P(8 correct) + P(9 correct) + P(10 correct)