Question

Multiple choice. You take a quiz with 6 multiple choice questions. After you studied, you estimated that you would have about an \(80 \%\) chance of getting any individual question right. What are your chances of getting them all right? Use at least 20 trials.

Short answer

The probability is 26.2%.

Step-by-step solution

Understand the Problem

We need to find the probability of correctly answering all 6 multiple-choice questions, where each question has an 80% chance of being answered correctly.

Set Up the Problem

Since each question has an independent probability of 0.8 of being correct, the probability of getting all 6 questions right is the product of the probabilities for each question.

Calculate the Probability

The probability of getting all questions correct is given by multiplying 0.8 (the probability of getting one question correct) by itself six times. The formula for this is: \[ P( ext{all correct}) = 0.8^6 \]

Compute the Result

Calculate \( 0.8^6 \) using a calculator or computation: \[ 0.8^6 = 0.262144 \]

Evaluate Solution

With this calculation, the probability of getting all questions correct is approximately 0.262, or 26.2%.

Key concepts

Independent Events

In probability, independent events are scenarios where the outcome of one event does not affect the outcome of another. This concept is essential when calculating probabilities over multiple trials, like in the case of a quiz with multiple choice questions.

Imagine you are tossing a coin. Whether it lands on heads or tails on the first toss does not influence what will happen on the second toss. Similarly, when you're taking a quiz, the probability of getting each question correct remains constant, regardless of your performance on other questions.

Understanding this concept is crucial because it allows us to simplify probability calculations by treating each event independently. In the case of our quiz with 6 questions, each question is an independent event with a consistent probability of being answered correctly.

Multiple Choice Questions

Multiple choice questions are a common form of assessment and they offer several potential answers, with only one correct option. In the context of probability, what makes multiple choice questions interesting is the fixed probability of getting the question right if you know the answer, or if you were guessing completely.

When you estimate your chance of answering correctly, like 80% in this case, it means that based on your knowledge and preparation, you are confident in choosing the correct answer 8 out of 10 times. This estimation plays a crucial role in determining the overall probability of passing or failing a quiz, especially if you can anticipate your performance across multiple questions.

Therefore, multiple choice questions are not just about luck; they are a blend of understanding and probability, which leads to strategic preparation and calculation.

Probability Calculation

Probability calculation is the process of determining how likely an event is to occur. It involves the use of formulas to calculate the chance of a specific outcome when certain conditions are met.

In the exercise given, we calculated the probability of answering all 6 questions correctly. Since each question is an independent event with a success probability of 0.8, the overall probability of getting all answers correct is the product of the individual probabilities.

This is mathematically represented as:

• Probability per question = 0.8
• Number of questions = 6
• Overall Probability = \( 0.8^6 \)

To find \( 0.8^6 \), you multiply 0.8 by itself six times, resulting in approximately 0.262, or 26.2%.

This probability calculation helps you understand how likely it is to achieve a full score on your quiz based on your estimated probability of success on each individual question.