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A box contains 8 marbles: 3 red, 4 blue, and 1 green. We draw without replacement from the box 4 times. Let X be the number of red marbles drawn out. Is X a binomial random variable? 1) Yes. 2) No. 3) Are we done yet? ;-) This is not the answer!

A box contains 8 marbles: 3 red, 4 blue, and 1 green. We draw without replacement from the box 4 times. Let X be the number of red marbles drawn out. Is X a binomial random variable? 1) Yes. 2) No. 3) Are we done yet? ;-) This is not the answer!

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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### Probability and Statistics Quiz **Question:** A box contains 8 marbles: 3 red, 4 blue, and 1 green. We draw without replacement from the box 4 times. Let \(X\) be the number of red marbles drawn out. Is \(X\) a binomial random variable? **Answer Options:** 1. Yes. 2. No. 3. Are we done yet? ;-) This is not the answer! ### Explanation: This question explores the concept of a binomial random variable in probability and statistics. For a variable to be binomial, the following conditions must be met: 1. **Fixed Number of Trials:** The experiment is repeated for a fixed number of trials. 2. **Two Possible Outcomes:** Each trial has exactly two outcomes – success and failure. 3. **Constant Probability:** The probability of success is the same for each trial. 4. **Independent Trials:** Each trial is independent of the others. When drawing marbles without replacement, the trials are not independent since the probability changes after each draw. Hence, in this scenario, \(X\) cannot be modeled as a binomial random variable even though it initially might seem to fit the binomial criteria. ### Detailed Analysis: - **Fixed Number of Trials:** Yes, there are 4 draws. - **Two Possible Outcomes:** If success is defined as drawing a red marble, then yes, each draw results in either a red marble (success) or not (failure). - **Constant Probability:** No, because once a marble is drawn, it is not replaced, altering the probability of drawing another red marble. - **Independent Trials:** No, because the draws affect each other. So, the correct answer is: **2) No.** (Note: Option 3 is a playful option and not a valid answer.)
Transcribed Image Text:### Probability and Statistics Quiz **Question:** A box contains 8 marbles: 3 red, 4 blue, and 1 green. We draw without replacement from the box 4 times. Let \(X\) be the number of red marbles drawn out. Is \(X\) a binomial random variable? **Answer Options:** 1. Yes. 2. No. 3. Are we done yet? ;-) This is not the answer! ### Explanation: This question explores the concept of a binomial random variable in probability and statistics. For a variable to be binomial, the following conditions must be met: 1. **Fixed Number of Trials:** The experiment is repeated for a fixed number of trials. 2. **Two Possible Outcomes:** Each trial has exactly two outcomes – success and failure. 3. **Constant Probability:** The probability of success is the same for each trial. 4. **Independent Trials:** Each trial is independent of the others. When drawing marbles without replacement, the trials are not independent since the probability changes after each draw. Hence, in this scenario, \(X\) cannot be modeled as a binomial random variable even though it initially might seem to fit the binomial criteria. ### Detailed Analysis: - **Fixed Number of Trials:** Yes, there are 4 draws. - **Two Possible Outcomes:** If success is defined as drawing a red marble, then yes, each draw results in either a red marble (success) or not (failure). - **Constant Probability:** No, because once a marble is drawn, it is not replaced, altering the probability of drawing another red marble. - **Independent Trials:** No, because the draws affect each other. So, the correct answer is: **2) No.** (Note: Option 3 is a playful option and not a valid answer.)
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