Probability Of One Correct Answer On A Multiple Choice Quiz

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In this article, we will delve into a probability problem involving a multiple-choice quiz. The scenario presents a situation where Van guesses on all eight questions, each having four answer choices. Our objective is to determine the probability that Van gets exactly one question correct. This problem falls under the realm of binomial probability, where we have a fixed number of independent trials, each with two possible outcomes: success (correct answer) or failure (incorrect answer). We will explore the underlying concepts, apply the binomial probability formula, and calculate the desired probability, rounding the final answer to the nearest thousandth. Understanding binomial probability is crucial in various fields, including statistics, data analysis, and decision-making, making this problem a valuable exercise in applying these principles.

To begin, let's dissect the problem statement. Van is taking a multiple-choice quiz with eight questions, and each question has four possible answer choices. Since Van is guessing on every question, the probability of him getting a question correct is 1/4, while the probability of him getting a question incorrect is 3/4. We are interested in finding the probability that Van gets exactly one question correct out of the eight. This is a classic example of a binomial probability problem because we have a fixed number of trials (eight questions), each trial is independent (Van's answer to one question doesn't affect his answers to other questions), there are only two possible outcomes for each trial (correct or incorrect), and the probability of success (getting a question correct) is constant (1/4) for each trial.

The key parameters in this problem are the number of trials (n = 8), the probability of success on a single trial (p = 1/4), and the number of successes we are interested in (k = 1). We need to use the binomial probability formula to calculate the probability of getting exactly one question correct. The binomial probability formula is given by:

P(k successes) = (n choose k) * p^k * (1 - p)^(n - k)

Where:

  • P(k successes) is the probability of getting exactly k successes in n trials.
  • (n choose k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials. It is calculated as n! / (k! * (n - k)!).
  • p is the probability of success on a single trial.
  • (1 - p) is the probability of failure on a single trial.
  • n is the total number of trials.
  • k is the number of successes we are interested in.

By understanding these parameters and the binomial probability formula, we can proceed to calculate the probability of Van getting exactly one question correct.

Now that we have a clear understanding of the problem and the binomial probability formula, let's apply it to calculate the probability of Van getting exactly one question correct. We have the following parameters:

  • n = 8 (number of questions)
  • k = 1 (number of correct answers)
  • p = 1/4 (probability of getting a question correct)
  • (1 - p) = 3/4 (probability of getting a question incorrect)

First, we need to calculate the binomial coefficient (n choose k), which represents the number of ways to choose 1 correct answer from 8 questions. Using the formula:

(n choose k) = n! / (k! * (n - k)!)

We have:

(8 choose 1) = 8! / (1! * 7!) = 8

This means there are 8 different ways Van can get exactly one question correct out of the eight.

Next, we need to calculate the probability of getting exactly one question correct and seven questions incorrect. This is given by:

p^k * (1 - p)^(n - k) = (1/4)^1 * (3/4)^7

Calculating this value, we get:

(1/4)^1 * (3/4)^7 = (1/4) * (2187/16384) = 2187/65536

Finally, we multiply the binomial coefficient by the probability we just calculated to get the probability of Van getting exactly one question correct:

P(1 success) = (8 choose 1) * (1/4)^1 * (3/4)^7 = 8 * (2187/65536) = 17496/65536

Now, we need to simplify this fraction and round the result to the nearest thousandth.

We have calculated the probability of Van getting exactly one question correct as:

P(1 success) = 17496/65536

To find the decimal value, we divide 17496 by 65536:

17496 / 65536 ≈ 0.267

Now, we round this result to the nearest thousandth, which means we need to keep three decimal places. In this case, 0.267 is already rounded to the nearest thousandth. Therefore, the probability that Van got exactly one question correct is approximately 0.267.

The probability of 0.267 indicates that there is a 26.7% chance that Van will get exactly one question correct if he guesses randomly on all eight questions. This result highlights the role of chance in multiple-choice quizzes and demonstrates how probability can be used to quantify the likelihood of specific outcomes. While a 26.7% chance might seem relatively low, it is significantly higher than the probability of getting all questions wrong or all questions correct, which would be much lower due to the nature of binomial distribution.

In conclusion, by applying the binomial probability formula, we determined that the probability of Van getting exactly one question correct on the multiple-choice quiz is approximately 0.267, or 26.7%. This problem illustrates the practical application of binomial probability in real-world scenarios and reinforces the importance of understanding probability concepts in various fields. The step-by-step approach, from understanding the problem to applying the formula and interpreting the result, provides a comprehensive understanding of how to solve such probability problems. This knowledge is not only useful in academic settings but also in everyday decision-making where assessing probabilities and risks is crucial.