Math Problem Statement

On a 5-question multiple-choice test, where each question has 4 answers, what would be the probability of getting at least one question wrong?

Solution

To solve this problem, we can use the complementary probability approach. Here's a breakdown of the steps:

Problem Breakdown:

  • Test setup: 5 multiple-choice questions, each with 4 answer choices.
  • We need to find the probability of getting at least one question wrong.

Complementary Probability Approach:

Instead of directly calculating the probability of getting at least one question wrong, we can first calculate the probability of getting all the questions correct, and then subtract this from 1.

  1. Probability of getting one question correct: Since each question has 4 options and only one is correct, the probability of choosing the correct answer is: P(correct answer)=14.P(\text{correct answer}) = \frac{1}{4}.

  2. Probability of getting all 5 questions correct: Since the questions are independent, the probability of getting all 5 questions correct is: P(all correct)=(14)5=11024.P(\text{all correct}) = \left( \frac{1}{4} \right)^5 = \frac{1}{1024}.

  3. Probability of getting at least one question wrong: This is the complement of getting all the questions correct, so we subtract the probability of getting all correct from 1: P(at least one wrong)=111024=102411024=10231024.P(\text{at least one wrong}) = 1 - \frac{1}{1024} = \frac{1024 - 1}{1024} = \frac{1023}{1024}.

Thus, the probability of getting at least one question wrong is 10231024\frac{1023}{1024}.

Would you like any further details or explanations on this? Here are some related questions for further exploration:

  1. What is the probability of getting exactly 1 question wrong?
  2. How does the probability change if the number of choices per question increases?
  3. What is the expected number of questions a person would get correct by guessing randomly?
  4. How would you calculate the probability of getting at least 2 questions wrong?
  5. Can the probability of getting at least one question wrong change if the test length is different?

Tip: When dealing with probabilities of "at least one" event, consider using the complement rule to simplify your calculation.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Probability
Complementary Probability
Independent Events

Formulas

P(correct answer) = 1/4
P(all correct) = (1/4)^5
P(at least one wrong) = 1 - P(all correct)

Theorems

Complement Rule

Suitable Grade Level

Grades 8-10