A student makes random guesses on 6 multiple-choice questions. One of the 5 choices in each question is correct. What is the probability that the student will get at most 4 answers right?
To find the probability of the student getting at most 4 answers right, we need to calculate the probability of the student getting 0, 1, 2, 3, or 4 answers right, and then sum them up.
Let's break down the problem step by step:
Step 1: Calculate the probability of the student getting exactly 0 answers right.
There is only one way to get all 6 answers wrong, so the probability is the probability of getting one answer wrong raised to the power of 6 (since there are 6 questions).
Probability of getting one answer wrong = 4/5
Probability of getting exactly 0 answers right = (4/5)^6
Step 2: Calculate the probability of the student getting exactly 1 answer right.
For each question, there is a 1/5 chance of getting it right and a 4/5 chance of getting it wrong. We need to choose 1 question to get right and 5 questions to get wrong.
Probability of getting exactly 1 answer right = (1/5) * (4/5)^5 * C(6,1), where C(6,1) is the number of ways to choose 1 question out of 6.
Step 3: Calculate the probability of the student getting exactly 2 answers right.
We need to choose 2 questions to get right and 4 questions to get wrong.
Probability of getting exactly 2 answers right = (1/5)^2 * (4/5)^4 * C(6,2)
Step 4: Calculate the probability of the student getting exactly 3 answers right.
We need to choose 3 questions to get right and 3 questions to get wrong.
Probability of getting exactly 3 answers right = (1/5)^3 * (4/5)^3 * C(6,3)
Step 5: Calculate the probability of the student getting exactly 4 answers right.
We need to choose 4 questions to get right and 2 questions to get wrong.
Probability of getting exactly 4 answers right = (1/5)^4 * (4/5)^2 * C(6,4)
Step 6: Add up the probabilities from steps 1 to 5 to get the probability of the student getting at most 4 answers right.
Probability of getting at most 4 answers right = Probability of getting exactly 0 answers right + Probability of getting exactly 1 answer right + Probability of getting exactly 2 answers right + Probability of getting exactly 3 answers right + Probability of getting exactly 4 answers right
I'll calculate the probabilities and sum them up to get the final result.