Understanding Random Sampling Without Replacement

Oct 13, 2025 by vinooalluri.com 122 views

Introduction

Imagine a scenario where five students are eager to play dodgeball, but they all want to be the team captain. To ensure fairness, the gym coach decides to use a random selection process to choose the team captain. In this article, we will delve into the world of probability and explore the concept of random sampling without replacement. We will examine the scenario where the coach selects two names at random from a hat containing the names of all five students.

The Problem

Five students, let's call them Alex, Ben, Charlie, David, and Emily, want to be the team captain of a dodgeball team. To resolve this issue, the gym coach decides to place the names of all five students into a hat and mix them well. The coach then selects two names at random without replacement. The two chosen names will be the team captains.

Understanding Random Sampling Without Replacement

Random sampling without replacement is a statistical technique where a subset of items is selected from a larger population, and the selected items are not replaced in the population. In this case, the gym coach is selecting two names from a hat containing five names. The coach will not replace the first name selected before choosing the second name.

Calculating the Probability of Each Student Being Chosen

To calculate the probability of each student being chosen as a team captain, we need to consider the total number of possible outcomes and the number of favorable outcomes for each student.

Let's assume that the hat contains the names of all five students: Alex, Ben, Charlie, David, and Emily. The coach will select two names at random without replacement.

Step 1: Calculate the Total Number of Possible Outcomes

The total number of possible outcomes can be calculated using the combination formula:

nCr = n! / (r!(n-r)!)

where n is the total number of items (5 students), r is the number of items to be chosen (2 team captains), and ! denotes the factorial function.

5C2 = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 × 4 × 3 × 2 × 1) / ((2 × 1)(3 × 2 × 1)) = 120 / (2 × 6) = 120 / 12 = 10

There are 10 possible outcomes when selecting two names from a hat containing five names.

Step 2: Calculate the Number of Favorable Outcomes for Each Student

To calculate the number of favorable outcomes for each student, we need to consider the number of ways each student can be chosen as one of the two team captains.

For example, if Alex is chosen as one of the team captains, there are 4 remaining students who can be chosen as the second team captain. The number of favorable outcomes for Alex is 4.

Similarly, the number of favorable outcomes for each student is:

Alex: 4
Ben: 4
Charlie: 4
David: 4
Emily: 4

Step 3: Calculate the Probability of Each Student Being Chosen

The probability of each student being chosen as a team captain can be calculated by dividing the number of favorable outcomes for each student by the total number of possible outcomes.

P(Alex) = 4/10 = 0.4 P(Ben) = 4/10 = 0.4 P(Charlie) = 4/10 = 0.4 P(David) = 4/10 = 0.4 P(Emily) = 4/10 = 0.4

Conclusion

In this article, we explored the concept of random sampling without replacement and applied it to a real-world scenario where five students want to be the team captain of a dodgeball team. We calculated the probability of each student being chosen as a team captain and found that each student has an equal probability of being chosen.

The Importance of Probability in Real-World Scenarios

Probability plays a crucial role in many real-world scenarios, including:

Insurance: Insurance companies use probability to calculate the likelihood of an event occurring and determine the premium to be paid.
Finance: Financial institutions use probability to calculate the risk of investments and determine the interest rates to be charged.
Medicine: Medical professionals use probability to diagnose diseases and determine the effectiveness of treatments.
Sports: Coaches and players use probability to make strategic decisions during games.

In conclusion, probability is a powerful tool that can be used to make informed decisions in a wide range of scenarios. By understanding the concept of random sampling without replacement, we can calculate the probability of each student being chosen as a team captain and make informed decisions in our personal and professional lives.

References

Khan Academy: Probability and Statistics
Wikipedia: Random Sampling
Math Is Fun: Probability

Introduction

In our previous article, we explored the concept of random sampling without replacement and applied it to a real-world scenario where five students want to be the team captain of a dodgeball team. We calculated the probability of each student being chosen as a team captain and found that each student has an equal probability of being chosen.

In this article, we will answer some frequently asked questions (FAQs) related to the concept of probability and random sampling without replacement.

Q&A

Q: What is the difference between random sampling with replacement and random sampling without replacement?

A: Random sampling with replacement involves selecting a subset of items from a larger population, replacing the selected items in the population, and repeating the process. Random sampling without replacement, on the other hand, involves selecting a subset of items from a larger population, not replacing the selected items in the population, and repeating the process.

Q: Why is random sampling without replacement used in many real-world scenarios?

A: Random sampling without replacement is used in many real-world scenarios because it provides a more accurate representation of the population. By not replacing the selected items in the population, we can ensure that each item has an equal chance of being selected.

Q: How do you calculate the probability of each student being chosen as a team captain?

A: To calculate the probability of each student being chosen as a team captain, we need to consider the total number of possible outcomes and the number of favorable outcomes for each student. We can use the combination formula to calculate the total number of possible outcomes and then divide the number of favorable outcomes for each student by the total number of possible outcomes.

Q: What is the probability of each student being chosen as a team captain in the scenario where five students want to be the team captain of a dodgeball team?

A: In the scenario where five students want to be the team captain of a dodgeball team, each student has an equal probability of being chosen as a team captain. The probability of each student being chosen is 4/10 or 0.4.

Q: Can you provide an example of how to calculate the probability of each student being chosen as a team captain?

A: Let's say we have five students: Alex, Ben, Charlie, David, and Emily. We want to calculate the probability of each student being chosen as a team captain. We can use the combination formula to calculate the total number of possible outcomes:

5C2 = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 × 4 × 3 × 2 × 1) / ((2 × 1)(3 × 2 × 1)) = 120 / (2 × 6) = 120 / 12 = 10

There are 10 possible outcomes when selecting two names from a hat containing five names. The number of favorable outcomes for each student is 4. Therefore, the probability of each student being chosen as a team captain is:

P(Alex) = 4/10 = 0.4 P(Ben) = 4/10 = 0.4 P(Charlie) = 4/10 = 0.4 P(David) = 4/10 = 0.4 P(Emily) = 4/10 = 0.4

Q: What is the importance of probability in real-world scenarios?

A: Probability plays a crucial role in many real-world scenarios, including insurance, finance, medicine, and sports. By understanding the concept of probability, we can make informed decisions in our personal and professional lives.

Q: Can you provide some examples of how probability is used in real-world scenarios?

A: Here are some examples of how probability is used in real-world scenarios:

Insurance: Insurance companies use probability to calculate the likelihood of an event occurring and determine the premium to be paid.
Finance: Financial institutions use probability to calculate the risk of investments and determine the interest rates to be charged.
Medicine: Medical professionals use probability to diagnose diseases and determine the effectiveness of treatments.
Sports: Coaches and players use probability to make strategic decisions during games.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to the concept of probability and random sampling without replacement. We provided examples of how to calculate the probability of each student being chosen as a team captain and discussed the importance of probability in real-world scenarios.

References

Khan Academy: Probability and Statistics
Wikipedia: Random Sampling
Math Is Fun: Probability