The Table Shows The Probabilities That $x$ Students Enroll In A Statistics Class At A College, But The Probability For $x=24$ Is Missing.Enter The Missing Value In The Table, And Then Find The Expected Value For The Number Of

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Introduction

In probability theory, the expected value is a measure of the central tendency of a random variable. It is calculated by multiplying each possible value of the random variable by its probability and summing the results. In this article, we will explore how to find the missing value in a table of probabilities and then calculate the expected value for the number of students enrolled in a statistics class.

The Table of Probabilities

x P(x)
20 0.05
21 0.10
22 0.15
23 0.20
24 ?
25 0.15
26 0.10
27 0.05

Finding the Missing Value

To find the missing value in the table, we can use the fact that the sum of the probabilities of all possible values of a random variable is equal to 1. In this case, we have:

0.05 + 0.10 + 0.15 + 0.20 + P(24) + 0.15 + 0.10 + 0.05 = 1

Simplifying the equation, we get:

0.80 + P(24) = 1

Subtracting 0.80 from both sides, we get:

P(24) = 0.20

Therefore, the missing value in the table is 0.20.

Calculating the Expected Value

Now that we have found the missing value in the table, we can calculate the expected value for the number of students enrolled in the statistics class. The expected value is calculated by multiplying each possible value of the random variable by its probability and summing the results. In this case, we have:

E(x) = 20(0.05) + 21(0.10) + 22(0.15) + 23(0.20) + 24(0.20) + 25(0.15) + 26(0.10) + 27(0.05)

E(x) = 1 + 2.1 + 3.3 + 4.6 + 4.8 + 3.75 + 2.6 + 1.35

E(x) = 24.4

Therefore, the expected value for the number of students enrolled in the statistics class is 24.4.

Conclusion

In this article, we have explored how to find the missing value in a table of probabilities and then calculate the expected value for the number of students enrolled in a statistics class. We have used the fact that the sum of the probabilities of all possible values of a random variable is equal to 1 to find the missing value, and then calculated the expected value by multiplying each possible value of the random variable by its probability and summing the results. The expected value is a measure of the central tendency of a random variable and is an important concept in probability theory.

Real-World Applications

The concept of expected value has many real-world applications. For example, in finance, the expected value of a stock's return is used to determine its value. In insurance, the expected value of a claim is used to determine the premium. In medicine, the expected value of a treatment's outcome is used to determine its effectiveness.

Future Research Directions

There are many future research directions in the area of expected value. For example, researchers could explore the use of expected value in machine learning algorithms. They could also investigate the use of expected value in decision-making under uncertainty.

References

  • [1] Ross, S. M. (2010). A First Course in Probability. 8th ed. Upper Saddle River, NJ: Prentice Hall.
  • [2] Sheldon, R. (2015). Probability and Statistics for Engineers and Scientists. 9th ed. Upper Saddle River, NJ: Prentice Hall.

Appendix

The following is a list of the formulas used in this article:

  • Expected Value: E(x) = ∑xP(x)
  • Sum of Probabilities: ∑P(x) = 1

Introduction

In our previous article, we explored how to find the missing value in a table of probabilities and then calculate the expected value for the number of students enrolled in a statistics class. In this article, we will answer some frequently asked questions (FAQs) related to the topic.

Q: What is the expected value?

A: The expected value is a measure of the central tendency of a random variable. It is calculated by multiplying each possible value of the random variable by its probability and summing the results.

Q: How do I find the missing value in a table of probabilities?

A: To find the missing value in a table of probabilities, you can use the fact that the sum of the probabilities of all possible values of a random variable is equal to 1. You can set up an equation using the given probabilities and solve for the missing value.

Q: What is the formula for the expected value?

A: The formula for the expected value is:

E(x) = ∑xP(x)

where x is the value of the random variable and P(x) is the probability of that value.

Q: How do I calculate the expected value?

A: To calculate the expected value, you need to multiply each possible value of the random variable by its probability and sum the results. You can use the formula:

E(x) = 20(0.05) + 21(0.10) + 22(0.15) + 23(0.20) + 24(0.20) + 25(0.15) + 26(0.10) + 27(0.05)

Q: What is the difference between the expected value and the mean?

A: The expected value and the mean are both measures of the central tendency of a random variable. However, the expected value is calculated using the probabilities of the random variable, while the mean is calculated using the values of the random variable.

Q: Can I use the expected value to make decisions?

A: Yes, the expected value can be used to make decisions. For example, in finance, the expected value of a stock's return is used to determine its value. In insurance, the expected value of a claim is used to determine the premium.

Q: What are some real-world applications of the expected value?

A: The expected value has many real-world applications, including:

  • Finance: The expected value of a stock's return is used to determine its value.
  • Insurance: The expected value of a claim is used to determine the premium.
  • Medicine: The expected value of a treatment's outcome is used to determine its effectiveness.

Q: What are some future research directions in the area of expected value?

A: Some future research directions in the area of expected value include:

  • Using the expected value in machine learning algorithms.
  • Investigating the use of the expected value in decision-making under uncertainty.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to the topic of the expected value. We have discussed how to find the missing value in a table of probabilities, how to calculate the expected value, and how to use the expected value in real-world applications. We have also discussed some future research directions in the area of expected value.

References

  • [1] Ross, S. M. (2010). A First Course in Probability. 8th ed. Upper Saddle River, NJ: Prentice Hall.
  • [2] Sheldon, R. (2015). Probability and Statistics for Engineers and Scientists. 9th ed. Upper Saddle River, NJ: Prentice Hall.

Appendix

The following is a list of the formulas used in this article:

  • Expected Value: E(x) = ∑xP(x)
  • Sum of Probabilities: ∑P(x) = 1

Note: The formulas are in LaTeX format.