Van Guessed On All 8 Questions Of A Multiple-choice Quiz. Each Question Has 4 Answer Choices. What Is The Probability That He Got Exactly 1 Question Correct? Round The Answer To The Nearest Thousandth.$[ \begin{array}{c} P(k \text{ Successes }) =
Introduction
In a multiple-choice quiz with 8 questions, each having 4 answer choices, Van surprisingly guessed all 8 questions correctly. However, we are interested in finding the probability that he got exactly 1 question correct. This problem can be approached using the concept of binomial probability, which is a fundamental concept in mathematics.
Understanding Binomial Probability
The binomial probability formula is used to calculate the probability of achieving 'k' successes in 'n' trials, where each trial has a probability 'p' of success. The formula is given by:
P(k successes) = (nCk) * (p^k) * ((1-p)^(n-k))
where nCk is the number of combinations of 'n' items taken 'k' at a time, p is the probability of success, and (1-p) is the probability of failure.
Calculating the Probability of Exactly 1 Correct Answer
In this problem, we have 8 questions (n=8), and we want to find the probability of getting exactly 1 question correct (k=1). Since each question has 4 answer choices, the probability of getting a correct answer (p) is 1/4, and the probability of getting an incorrect answer (1-p) is 3/4.
Using the binomial probability formula, we can calculate the probability of getting exactly 1 correct answer as follows:
P(1 success) = (8C1) * ((1/4)^1) * ((3/4)^(8-1)) = (8) * (1/4) * (3/4)^7 = 8 * (1/4) * (2187/16384) = 8 * (2187/65536) = 17496/65536 = 0.267
Rounding the Answer to the Nearest Thousandth
To round the answer to the nearest thousandth, we need to look at the thousandth place value, which is 6 in this case. Since the digit in the thousandth place is less than 5, we round down to 0.267.
Conclusion
In conclusion, the probability that Van got exactly 1 question correct in the multiple-choice quiz is approximately 0.267, or 26.7%. This result highlights the importance of understanding binomial probability and its applications in real-world scenarios.
Discussion
This problem can be extended to find the probability of getting exactly 2, 3, 4, 5, 6, 7, or 8 correct answers. By using the binomial probability formula, we can calculate the probability of each scenario and compare the results.
Calculating the Probability of Exactly 2 Correct Answers
Using the binomial probability formula, we can calculate the probability of getting exactly 2 correct answers as follows:
P(2 successes) = (8C2) * ((1/4)^2) * ((3/4)^(8-2)) = (28) * (1/16) * (3/4)^6 = 28 * (1/16) * (729/4096) = 28 * (729/65536) = 20412/65536 = 0.311
Calculating the Probability of Exactly 3 Correct Answers
Using the binomial probability formula, we can calculate the probability of getting exactly 3 correct answers as follows:
P(3 successes) = (8C3) * ((1/4)^3) * ((3/4)^(8-3)) = (56) * (1/64) * (3/4)^5 = 56 * (1/64) * (243/1024) = 56 * (243/65536) = 13608/65536 = 0.207
Calculating the Probability of Exactly 4 Correct Answers
Using the binomial probability formula, we can calculate the probability of getting exactly 4 correct answers as follows:
P(4 successes) = (8C4) * ((1/4)^4) * ((3/4)^(8-4)) = (70) * (1/256) * (3/4)^4 = 70 * (1/256) * (81/256) = 70 * (81/65536) = 5670/65536 = 0.087
Calculating the Probability of Exactly 5 Correct Answers
Using the binomial probability formula, we can calculate the probability of getting exactly 5 correct answers as follows:
P(5 successes) = (8C5) * ((1/4)^5) * ((3/4)^(8-5)) = (56) * (1/1024) * (3/4)^3 = 56 * (1/1024) * (27/64) = 56 * (27/65536) = 1512/65536 = 0.023
Calculating the Probability of Exactly 6 Correct Answers
Using the binomial probability formula, we can calculate the probability of getting exactly 6 correct answers as follows:
P(6 successes) = (8C6) * ((1/4)^6) * ((3/4)^(8-6)) = (28) * (1/4096) * (3/4)^2 = 28 * (1/4096) * (9/16) = 28 * (9/65536) = 252/65536 = 0.004
Calculating the Probability of Exactly 7 Correct Answers
Using the binomial probability formula, we can calculate the probability of getting exactly 7 correct answers as follows:
P(7 successes) = (8C7) * ((1/4)^7) * ((3/4)^(8-7)) = (8) * (1/16384) * (3/4) = 8 * (3/65536) = 24/65536 = 0.0004
Calculating the Probability of Exactly 8 Correct Answers
Using the binomial probability formula, we can calculate the probability of getting exactly 8 correct answers as follows:
P(8 successes) = (8C8) * ((1/4)^8) * ((3/4)^(8-8)) = (1) * (1/262144) * (1) = 1/262144 = 0.0000038
Conclusion
Q: What is the probability of getting exactly 1 correct answer in a multiple-choice quiz with 8 questions, each having 4 answer choices?
A: The probability of getting exactly 1 correct answer is approximately 0.267, or 26.7%.
Q: How did you calculate the probability of getting exactly 1 correct answer?
A: We used the binomial probability formula, which is given by:
P(k successes) = (nCk) * (p^k) * ((1-p)^(n-k))
where nCk is the number of combinations of 'n' items taken 'k' at a time, p is the probability of success, and (1-p) is the probability of failure.
Q: What is the probability of getting exactly 2 correct answers in a multiple-choice quiz with 8 questions, each having 4 answer choices?
A: The probability of getting exactly 2 correct answers is approximately 0.311.
Q: How did you calculate the probability of getting exactly 2 correct answers?
A: We used the binomial probability formula, which is given by:
P(k successes) = (nCk) * (p^k) * ((1-p)^(n-k))
where nCk is the number of combinations of 'n' items taken 'k' at a time, p is the probability of success, and (1-p) is the probability of failure.
Q: What is the probability of getting exactly 3 correct answers in a multiple-choice quiz with 8 questions, each having 4 answer choices?
A: The probability of getting exactly 3 correct answers is approximately 0.207.
Q: How did you calculate the probability of getting exactly 3 correct answers?
A: We used the binomial probability formula, which is given by:
P(k successes) = (nCk) * (p^k) * ((1-p)^(n-k))
where nCk is the number of combinations of 'n' items taken 'k' at a time, p is the probability of success, and (1-p) is the probability of failure.
Q: What is the probability of getting exactly 4 correct answers in a multiple-choice quiz with 8 questions, each having 4 answer choices?
A: The probability of getting exactly 4 correct answers is approximately 0.087.
Q: How did you calculate the probability of getting exactly 4 correct answers?
A: We used the binomial probability formula, which is given by:
P(k successes) = (nCk) * (p^k) * ((1-p)^(n-k))
where nCk is the number of combinations of 'n' items taken 'k' at a time, p is the probability of success, and (1-p) is the probability of failure.
Q: What is the probability of getting exactly 5 correct answers in a multiple-choice quiz with 8 questions, each having 4 answer choices?
A: The probability of getting exactly 5 correct answers is approximately 0.023.
Q: How did you calculate the probability of getting exactly 5 correct answers?
A: We used the binomial probability formula, which is given by:
P(k successes) = (nCk) * (p^k) * ((1-p)^(n-k))
where nCk is the number of combinations of 'n' items taken 'k' at a time, p is the probability of success, and (1-p) is the probability of failure.
Q: What is the probability of getting exactly 6 correct answers in a multiple-choice quiz with 8 questions, each having 4 answer choices?
A: The probability of getting exactly 6 correct answers is approximately 0.004.
Q: How did you calculate the probability of getting exactly 6 correct answers?
A: We used the binomial probability formula, which is given by:
P(k successes) = (nCk) * (p^k) * ((1-p)^(n-k))
where nCk is the number of combinations of 'n' items taken 'k' at a time, p is the probability of success, and (1-p) is the probability of failure.
Q: What is the probability of getting exactly 7 correct answers in a multiple-choice quiz with 8 questions, each having 4 answer choices?
A: The probability of getting exactly 7 correct answers is approximately 0.0004.
Q: How did you calculate the probability of getting exactly 7 correct answers?
A: We used the binomial probability formula, which is given by:
P(k successes) = (nCk) * (p^k) * ((1-p)^(n-k))
where nCk is the number of combinations of 'n' items taken 'k' at a time, p is the probability of success, and (1-p) is the probability of failure.
Q: What is the probability of getting exactly 8 correct answers in a multiple-choice quiz with 8 questions, each having 4 answer choices?
A: The probability of getting exactly 8 correct answers is approximately 0.0000038.
Q: How did you calculate the probability of getting exactly 8 correct answers?
A: We used the binomial probability formula, which is given by:
P(k successes) = (nCk) * (p^k) * ((1-p)^(n-k))
where nCk is the number of combinations of 'n' items taken 'k' at a time, p is the probability of success, and (1-p) is the probability of failure.
Conclusion
In conclusion, the probabilities of getting exactly 1, 2, 3, 4, 5, 6, 7, or 8 correct answers in a multiple-choice quiz with 8 questions, each having 4 answer choices, are approximately 0.267, 0.311, 0.207, 0.087, 0.023, 0.004, 0.0004, and 0.0000038, respectively. These results highlight the importance of understanding binomial probability and its applications in real-world scenarios.