Astudent 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 that the student will get 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 add them together.
Let's start by finding the probability of the student getting exactly 0 answers right. Since there are 5 choices for each question and only 1 of them is correct, the probability of getting a single question wrong is 4/5. Since there are 6 questions, the probability of getting all 6 questions wrong is calculated as:
(4/5) * (4/5) * (4/5) * (4/5) * (4/5) * (4/5) = (4/5)^6
Next, let's find the probability of the student getting exactly 1 answer right. The student can choose any of the 6 questions to answer correctly, and the probability of answering it correctly is 1/5. The remaining 5 questions will be answered incorrectly with a probability of 4/5. So, the probability of getting exactly 1 answer right is:
6 * (1/5) * (4/5)^5
Similarly, we can calculate the probabilities of getting exactly 2, 3, and 4 answers right using the same logic. The formulas for these probabilities are:
- Probability of getting exactly 2 answers right: 15 * (1/5)^2 * (4/5)^4
- Probability of getting exactly 3 answers right: 20 * (1/5)^3 * (4/5)^3
- Probability of getting exactly 4 answers right:15 * (1/5)^4 * (4/5)^2
Finally, to find the probability of getting at most 4 answers right, we add up these probabilities:
(4/5)^6 + 6 * (1/5) * (4/5)^5 + 15 * (1/5)^2 * (4/5)^4 + 20 * (1/5)^3 * (4/5)^3 + 15 * (1/5)^4 * (4/5)^2