A multiple choice test has 10 questions each of which has 4 possible answers, only one of which is correct, if Judy, who forgot to study for the test, guesses on all questions, what is the probability that she will answer exactly 3 questions correctly? (hint: Binomial Distribution) 0.5006 0.0021 0.0156 0.2816 0.2503
A multiple choice test has 10 questions each of which has 4 possible answers, only one of which is correct, if Judy, who forgot to study for the test, guesses on all questions, what is the probability that she will answer exactly 3 questions correctly? (hint: Binomial Distribution) 0.5006 0.0021 0.0156 0.2816 0.2503
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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![### Educational Content on Probability: Understanding Binomial Distribution
**Problem Scenario:**
A multiple choice test consists of 10 questions, each with 4 possible answers, only one of which is correct. If Judy, who forgot to study for the test, guesses on all questions, we seek to find the probability that she will answer exactly 3 questions correctly. (Hint: Binomial Distribution)
**Answer Choices:**
- 0.5006
- 0.0021
- 0.0156
- 0.2816
- 0.2503
### Explanation of the Concept:
This problem involves calculating probabilities using the **binomial distribution**. The binomial distribution models the number of successful outcomes in a fixed number of independent and identically distributed binary experiments. Here, each question represents a binary experiment with a success (correct answer) happening with a probability of 0.25 (since 1 out of 4 options is correct).
**Steps to Solve:**
1. **Define the variables:**
- Number of trials (n) = 10 (the number of questions)
- Probability of success on a single trial (p) = 0.25
- Number of successes (k) we are interested in = 3
2. **Binomial Probability Formula:**
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
where \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).
By plugging in the values, you can calculate the probability of exactly 3 correct answers:
\[
P(X = 3) = \binom{10}{3} (0.25)^3 (0.75)^7
\]
This setup helps in reinforcing the understanding of how to use the binomial distribution in probability scenarios.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5247ce5-a3e5-4e1f-ba29-c615e03ea9bd%2F24d20306-c204-420a-9569-8b1da23b974e%2F8c4g5pv_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Content on Probability: Understanding Binomial Distribution
**Problem Scenario:**
A multiple choice test consists of 10 questions, each with 4 possible answers, only one of which is correct. If Judy, who forgot to study for the test, guesses on all questions, we seek to find the probability that she will answer exactly 3 questions correctly. (Hint: Binomial Distribution)
**Answer Choices:**
- 0.5006
- 0.0021
- 0.0156
- 0.2816
- 0.2503
### Explanation of the Concept:
This problem involves calculating probabilities using the **binomial distribution**. The binomial distribution models the number of successful outcomes in a fixed number of independent and identically distributed binary experiments. Here, each question represents a binary experiment with a success (correct answer) happening with a probability of 0.25 (since 1 out of 4 options is correct).
**Steps to Solve:**
1. **Define the variables:**
- Number of trials (n) = 10 (the number of questions)
- Probability of success on a single trial (p) = 0.25
- Number of successes (k) we are interested in = 3
2. **Binomial Probability Formula:**
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
where \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).
By plugging in the values, you can calculate the probability of exactly 3 correct answers:
\[
P(X = 3) = \binom{10}{3} (0.25)^3 (0.75)^7
\]
This setup helps in reinforcing the understanding of how to use the binomial distribution in probability scenarios.
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