cotall My Courses CRN 40938 Chapter 4 HW - Discrete Random Variable Current score: 82.5/100 points Submit aselgnmant *Al Category: Homework At The Fencing Center, $$ 59 \% $$ of the fencers use the foil as their main weapon. We randomly survey 18 fencers at The Fencing Center. We are interested in the number of fencers who do not use the foil as their main weapon. (a) Which of the following defines the random variable $$ X $$ ? A $$ X= $$ The probability that a randomly selected fencer doesn't use foil as their main weapon. B $$ X= $$ The probability that a randomly selected fencer uses foil as their main weapon. C $$ X= $$ The number of fencers who do not use the foil as their main weapon. D $$ X= $$ The number of fencers who participated in the survey. (b) List all the values that $$ X $$ may take on. $$ x= $$ $$ \square $$ Enter your answer (Enter your answer as a comma-separated list.) (c) Give the distribution of $$ X $$. $$ X \sim $$ $$ \square $$ Enter your answer Enter your answer $$ \square $$ , Enter your answer ) (Enter numerical values as integers or decimals.)

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cotall My Courses > CRN 40938 Chapter 4 HW - Discrete Random Variable Current score: 82.5/100 points Submit aselgnmant *Al Category: Homework At The Fencing Center, $$ 59 \% $$ of the fencers use the foil as their main weapon. We randomly survey 18 fencers at The Fencing Center. We are interested in the number of fencers who do not use the foil as their main weapon. (a) Which of the following defines the random variable $$ X $$ ? A $$ X= $$ The probability that a randomly selected fencer doesn't use foil as their main weapon. B $$ X= $$ The probability that a randomly selected fencer uses foil as their main weapon. C $$ X= $$ The number of fencers who do not use the foil as their main weapon. D $$ X= $$ The number of fencers who participated in the survey. (b) List all the values that $$ X $$ may take on. $$ x= $$ $$ \square $$ Enter your answer (Enter your answer as a comma-separated list.) (c) Give the distribution of $$ X $$. $$ X \sim $$ $$ \square $$ Enter your answer Enter your answer $$ \square $$ , Enter your answer ) (Enter numerical values as integers or decimals.)

UpStudy AI · Accepted Answer

Sure, I'd be happy to help you with your homework assignment on discrete random variables. Let's go through each part of the problem step by step.

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### **Problem Statement Recap**

At The Fencing Center, **59%** of the fencers use the **foil** as their main weapon. You randomly survey **18 fencers** and are interested in the number of fencers who **do not** use the foil as their main weapon.

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### **Part (a): Defining the Random Variable $$ X $$**

**Question:**
Which of the following defines the random variable $$ X $$?

**Options:**
- **A.** $$ X = $$ The probability that a randomly selected fencer doesn't use foil as their main weapon.
- **B.** $$ X = $$ The probability that a randomly selected fencer uses foil as their main weapon.
- **C.** $$ X = $$ The number of fencers who do not use the foil as their main weapon.
- **D.** $$ X = $$ The number of fencers who participated in the survey.

**Answer:**
**C.** $$ X = $$ The number of fencers who do not use the foil as their main weapon.

**Explanation:**
The random variable $$ X $$ is defined as the count of fencers who **do not** use the foil as their main weapon out of the 18 surveyed. Option C accurately captures this definition.

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### **Part (b): Listing All Possible Values of $$ X $$**

**Question:**
List all the values that $$ X $$ may take on.

**Answer:**
$$ x = 0, 1, 2, 3, \ldots, 18 $$

**Explanation:**
Since you're surveying 18 fencers, the number of fencers who do not use the foil can range from **0** (all fencers use the foil) to **18** (none of the fencers use the foil). Therefore, $$ X $$ can take any integer value between 0 and 18, inclusive.

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### **Part (c): Determining the Distribution of $$ X $$**

**Question:**
Give the distribution of $$ X $$.

**Answer:**
$$ X \sim \text{Binomial}(n = 18, p = 0.41) $$

**Explanation:**
- **Binomial Distribution:** This distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success.
  
- **Parameters:**
  - $$ n = 18 $$: The number of trials (fencers surveyed).
  - $$ p = 0.41 $$: The probability of "success," where success is defined as a fencer **not** using the foil as their main weapon. Since 59% use the foil, the probability of not using it is $$ 1 - 0.59 = 0.41 $$.

Therefore, the random variable $$ X $$ follows a binomial distribution with parameters $$ n = 18 $$ and $$ p = 0.41 $$.

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### **Summary of Answers**

- **(a)** **C**
  
- **(b)** $$ x = 0, 1, 2, \ldots, 18 $$
  
- **(c)** $$ X \sim \text{Binomial}(18, 0.41) $$

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Feel free to let me know if you need further clarification on any of these parts!
**Part (a):** $$ X $$ is the number of fencers who do not use the foil as their main weapon. **Part (b):** $$ x = 0, 1, 2, \ldots, 18 $$ **Part (c):** $$ X \sim \text{Binomial}(18, 0.41) $$

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