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Chapter 6: Problem 38

A quiz in a statistics course has four multiple-choice questions, each with five possible answers. A passing grade is three or more correct answers to the four questions. Allison has not studied for the quiz. She has no idea of the correct answer to any of the questions and decides to guess at random for each. a. Find the probability she lucks out and answers all four questions correctly. b. Find the probability that she passes the quiz.

Short Answer

Expert verified
a. Probability of 4 correct: 0.0016. b. Probability of passing: 0.0272.

Step by step solution

01

Define the Successful Trial

Since each question has five possible answers, the probability of guessing a single question correctly is \( p = \frac{1}{5} = 0.2 \).
02

Set Up the Binomial Distribution

This scenario can be modeled using a binomial distribution where the number of questions \( n = 4 \), and the probability of success on a single question \( p = 0.2 \). The random variable \( X \), representing the number of correct answers, follows a binomial distribution \( X \sim B(4, 0.2) \).
03

Calculate Probability for 4 Correct Answers

To find \( P(X = 4) \), we use the binomial probability formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]For all answers correct (\( k = 4 \)):\[ P(X = 4) = \binom{4}{4} (0.2)^4 (0.8)^{0} = 1 \times 0.0016 \times 1 = 0.0016 \]
04

Find Probability of Passing the Quiz

Allison passes the quiz if she gets at least 3 correct. We need to find \( P(X \geq 3) \). This is calculated by finding \( P(X = 3) \) and \( P(X = 4) \), then summing them:\[ P(X = 3) = \binom{4}{3} \times (0.2)^3 \times (0.8)^{1} = 4 \times 0.008 \times 0.8 = 0.0256 \]Then, add the probabilities for 3 and 4 correct answers:\[ P(X \geq 3) = P(X = 3) + P(X = 4) = 0.0256 + 0.0016 = 0.0272 \]
05

Conclusion

The probability that Allison answers all four questions correctly is 0.0016, and the probability that she passes the quiz is 0.0272.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is the measure of the likelihood that a particular event will occur. In this scenario, Allison is guessing the answers to a quiz. Each question has multiple choices, making the probability of guessing correctly quite specific. Here, each question has five possible answers, meaning Allison has a 1 in 5 chance, or a probability of 0.2, to guess the answer correctly.
  • Probability values range from 0 to 1.
  • A probability of 0 means the event cannot happen.
  • A probability of 1 means the event is certain.
For Allison's quiz, the probabilities are part of a binomial distribution, where the outcomes are either a success (correct answer) or a failure (wrong answer). Understanding probability helps to determine the likelihood that she will pass, which in her case, involves random guessing.
Statistics Quiz
Statistics quizzes often use probability concepts to test understanding of different statistical theories. In the case of Allison's quiz, the challenge is whether random guessing can meet the statistical requirement to pass. The quiz consists of four multiple-choice questions, and to pass, Allison needs to get at least three questions right. This means she needs almost perfect guessing to succeed.
  • The probability model used: Binomial Distribution.
  • Number of trials (questions): 4.
  • Probability of success per trial: 0.2.
Using the binomial distribution, you can calculate the likelihood of Allison getting a set number of questions right, which is essential in determining her chances of passing the quiz. This statistical element adds depth to problem-solving in a quiz scenario.
Multiple-Choice Questions
Multiple-choice questions provide distinct probabilities with finite possible answers. In Allison's quiz, each question has five possible answers, making it a classic example of a probability problem. When guessing on such questions, you can analyze outcomes using specific methods.
  • Each question's success probability: 0.2, for guessing correctly.
  • Failure probability: 0.8, since there are more incorrect options.
The way multiple-choice questions are structured provides a clear way to apply the binomial probability formula. For instance, to get a specific number of questions correct, you multiply the success probability raised to the number of correct answers by the failure probability raised to the remaining attempts. This structured approach makes dealing with randomness more predictable and mathematical in nature.

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Most popular questions from this chapter

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