Johnni Guessed On All 8 Questions Of A Multiple-choice Quiz. Each Question Has 4 Answer Choices. What Is The Probability That She Got Exactly 3 Questions Correct? Round The Answer To The Nearest Thousandth.$[ P(k \text{ Successes }) =

Introduction

In this article, we will delve into the world of probability and explore the concept of calculating the probability of a specific outcome in a multiple-choice quiz. We will use the binomial probability formula to determine the likelihood of Johnni getting exactly 3 questions correct out of 8.

The Binomial Probability Formula

The binomial probability formula is used to calculate the probability of achieving 'k' successes in 'n' trials, where the probability of success in each trial is 'p'. The formula is given by:

${ P(k \text{ successes }) = \binom{n}{k} p^k (1-p)^{n-k} }$

where:

• ${ \binom{n}{k} }$ is the binomial coefficient, which represents the number of ways to choose 'k' items from a set of 'n' items.
• ${ p }$ is the probability of success in each trial.
• ${ n }$ is the total number of trials.
• ${ k }$ is the number of successes.

Calculating the Probability of Exactly 3 Correct Answers

In this problem, we have 8 questions (n = 8), and we want to find the probability of getting exactly 3 questions correct (k = 3). Since each question has 4 answer choices, the probability of getting a correct answer is 1/4, and the probability of getting an incorrect answer is 3/4.

Using the binomial probability formula, we can calculate the probability of exactly 3 correct answers as follows:

${ P(3 \text{ correct answers}) = \binom{8}{3} \left(\frac{1}{4}\right)^3 \left(\frac{3}{4}\right)^5 }$

Calculating the Binomial Coefficient

The binomial coefficient ${ \binom{8}{3} }$ can be calculated as follows:

${ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 }$

Calculating the Probability

Now that we have the binomial coefficient, we can calculate the probability of exactly 3 correct answers:

${ P(3 \text{ correct answers}) = 56 \left(\frac{1}{4}\right)^3 \left(\frac{3}{4}\right)^5 }$

${ P(3 \text{ correct answers}) = 56 \left(\frac{1}{64}\right) \left(\frac{243}{1024}\right) }$

${ P(3 \text{ correct answers}) = 56 \left(\frac{243}{65536}\right) }$

${ P(3 \text{ correct answers}) = \frac{13668}{65536} }$

Rounding the Answer to the Nearest Thousandth

To round the answer to the nearest thousandth, we can divide the numerator and denominator by 65536:

${ P(3 \text{ correct answers}) = \frac{13668}{65536} \approx 0.208 }$

Therefore, the probability of Johnni getting exactly 3 questions correct out of 8 is approximately 0.208, or 20.8%.

Conclusion

In this article, we used the binomial probability formula to calculate the probability of exactly 3 correct answers in a multiple-choice quiz with 8 questions. We found that the probability is approximately 0.208, or 20.8%. This result can be useful in understanding the likelihood of achieving a specific outcome in a multiple-choice quiz.

Introduction

In our previous article, we explored the concept of binomial probability and calculated the probability of exactly 3 correct answers in a multiple-choice quiz with 8 questions. In this article, we will address some frequently asked questions related to binomial probability.

Q: What is the binomial probability formula?

A: The binomial probability formula is used to calculate the probability of achieving 'k' successes in 'n' trials, where the probability of success in each trial is 'p'. The formula is given by:

${ P(k \text{ successes }) = \binom{n}{k} p^k (1-p)^{n-k} }$

Q: What is the binomial coefficient?

A: The binomial coefficient, denoted by ${ \binom{n}{k} }$, represents the number of ways to choose 'k' items from a set of 'n' items. It can be calculated as follows:

${ \binom{n}{k} = \frac{n!}{k!(n-k)!} }$

Q: What is the probability of success in each trial?

A: The probability of success in each trial is denoted by 'p'. It represents the likelihood of achieving a success in a single trial.

Q: What is the total number of trials?

A: The total number of trials is denoted by 'n'. It represents the number of trials in which the probability of success is 'p'.

Q: What is the number of successes?

A: The number of successes is denoted by 'k'. It represents the number of times the probability of success is achieved in the 'n' trials.

Q: How do I calculate the binomial probability?

A: To calculate the binomial probability, you need to follow these steps:

1. Calculate the binomial coefficient ${ \binom{n}{k} }$.
2. Calculate the probability of success in each trial, which is 'p'.
3. Calculate the probability of failure in each trial, which is ${ 1-p }$.
4. Calculate the probability of achieving 'k' successes in 'n' trials using the binomial probability formula.

Q: What is the significance of binomial probability in real-life scenarios?

A: Binomial probability has numerous applications in real-life scenarios, such as:

• Quality control: Binomial probability is used to determine the probability of achieving a certain quality level in a manufacturing process.
• Medical research: Binomial probability is used to determine the probability of a patient responding to a treatment.
• Finance: Binomial probability is used to determine the probability of achieving a certain return on investment.

Q: Can I use binomial probability to calculate the probability of more than one success?

A: Yes, you can use binomial probability to calculate the probability of more than one success. However, you need to use the binomial probability formula for each possible number of successes and then sum the probabilities to get the total probability.

Conclusion

In this article, we addressed some frequently asked questions related to binomial probability. We hope that this article has provided you with a better understanding of binomial probability and its applications in real-life scenarios.

Additional Resources

For more information on binomial probability, you can refer to the following resources:

• Binomial probability formula: ${ P(k \text{ successes }) = \binom{n}{k} p^k (1-p)^{n-k} }$
• Binomial coefficient: ${ \binom{n}{k} = \frac{n!}{k!(n-k)!} }$
• Binomial probability calculator: You can use an online binomial probability calculator to calculate the probability of achieving a certain number of successes in a given number of trials.

Final Thoughts

Binomial probability is a powerful tool for calculating the probability of achieving a certain number of successes in a given number of trials. It has numerous applications in real-life scenarios, and understanding binomial probability can help you make informed decisions in various fields.