A Hat Contains Slips Of Paper With The Names Of The 26 Other Students In Eduardo's Class, 10 Of Whom Are Boys. To Determine His Partners For The Group Project, Eduardo Has To Pull Two Names Out Of The Hat Without Replacing Them.What Is The Probability

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Introduction

In a class of 26 students, 10 of whom are boys, Eduardo has to determine his partners for the group project by pulling two names out of a hat without replacing them. This scenario presents a classic problem in probability, where we need to calculate the likelihood of drawing specific combinations of names from the hat. In this article, we will delve into the world of probability and explore the concept of combinations, permutations, and the probability of drawing specific pairs of names from the hat.

Understanding the Basics of Probability

Probability is a measure of the likelihood of an event occurring. It is often expressed as a value between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. In the context of this problem, we are interested in finding the probability of drawing two specific names from the hat without replacement.

Calculating Combinations

To calculate the probability of drawing two specific names from the hat, we need to first determine the total number of possible combinations of names that can be drawn. This can be calculated using the formula for combinations:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of items (in this case, 26 students), k is the number of items to be chosen (in this case, 2 names), and ! represents the factorial function (e.g., 5! = 5 × 4 × 3 × 2 × 1).

Calculating the Total Number of Possible Combinations

Using the formula for combinations, we can calculate the total number of possible combinations of names that can be drawn from the hat:

C(26, 2) = 26! / (2!(26-2)!) = 26! / (2!24!) = (26 × 25) / (2 × 1) = 325

Calculating the Probability of Drawing Specific Pairs of Names

Now that we have determined the total number of possible combinations of names that can be drawn, we can calculate the probability of drawing specific pairs of names from the hat. Since there are 325 possible combinations of names, and we are interested in finding the probability of drawing a specific pair of names, we can use the following formula:

P(event) = Number of favorable outcomes / Total number of possible outcomes

In this case, the number of favorable outcomes is 1 (since we are interested in finding the probability of drawing a specific pair of names), and the total number of possible outcomes is 325.

Calculating the Probability of Drawing a Specific Pair of Names

Using the formula for probability, we can calculate the probability of drawing a specific pair of names from the hat:

P(event) = 1 / 325 = 0.0030769

Calculating the Probability of Drawing Two Boys

Now that we have calculated the probability of drawing a specific pair of names from the hat, we can calculate the probability of drawing two boys. Since there are 10 boys in the class, the number of favorable outcomes is the number of ways to choose 2 boys from 10, which can be calculated using the formula for combinations:

C(10, 2) = 10! / (2!(10-2)!) = 10! / (2!8!) = (10 × 9) / (2 × 1) = 45

The total number of possible outcomes remains the same, which is 325. Therefore, the probability of drawing two boys can be calculated as follows:

P(event) = Number of favorable outcomes / Total number of possible outcomes = 45 / 325 = 0.13846154

Calculating the Probability of Drawing Two Girls

Similarly, we can calculate the probability of drawing two girls. Since there are 16 girls in the class, the number of favorable outcomes is the number of ways to choose 2 girls from 16, which can be calculated using the formula for combinations:

C(16, 2) = 16! / (2!(16-2)!) = 16! / (2!14!) = (16 × 15) / (2 × 1) = 120

The total number of possible outcomes remains the same, which is 325. Therefore, the probability of drawing two girls can be calculated as follows:

P(event) = Number of favorable outcomes / Total number of possible outcomes = 120 / 325 = 0.36923077

Conclusion

In this article, we have explored the concept of probability and its application to a real-world scenario. We have calculated the probability of drawing specific pairs of names from a hat without replacement, as well as the probability of drawing two boys or two girls. The results demonstrate the importance of considering the total number of possible outcomes when calculating probability, and the need to use the correct formula for combinations and permutations.

Final Thoughts

Probability is a fundamental concept in mathematics that has numerous applications in real-world scenarios. By understanding the basics of probability and how to calculate combinations and permutations, we can make informed decisions and predictions in a wide range of fields, from finance and economics to science and engineering. In the context of this problem, the probability of drawing specific pairs of names from a hat without replacement can be used to inform decisions about group project partners, and the probability of drawing two boys or two girls can be used to make predictions about the composition of the group.

References

  • [1] "Probability" by Khan Academy
  • [2] "Combinations and Permutations" by Math Is Fun
  • [3] "Probability and Statistics" by Coursera

Further Reading

  • [1] "Introduction to Probability" by Charles M. Grinstead and J. Laurie Snell
  • [2] "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole and Raymond H. Myers
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Introduction

In our previous article, we explored the concept of probability and its application to a real-world scenario. We calculated the probability of drawing specific pairs of names from a hat without replacement, as well as the probability of drawing two boys or two girls. In this article, we will address some of the most frequently asked questions related to this topic.

Q&A

Q1: What is the probability of drawing a specific pair of names from the hat?

A1: The probability of drawing a specific pair of names from the hat is 1/325, which is approximately 0.0030769.

Q2: How do I calculate the probability of drawing two boys from the hat?

A2: To calculate the probability of drawing two boys from the hat, you need to calculate the number of ways to choose 2 boys from 10, which is C(10, 2) = 45. Then, you divide this number by the total number of possible outcomes, which is 325. The probability of drawing two boys is 45/325, which is approximately 0.13846154.

Q3: What is the probability of drawing two girls from the hat?

A3: To calculate the probability of drawing two girls from the hat, you need to calculate the number of ways to choose 2 girls from 16, which is C(16, 2) = 120. Then, you divide this number by the total number of possible outcomes, which is 325. The probability of drawing two girls is 120/325, which is approximately 0.36923077.

Q4: How do I calculate the probability of drawing a specific pair of names from the hat without replacement?

A4: To calculate the probability of drawing a specific pair of names from the hat without replacement, you need to calculate the total number of possible combinations of names that can be drawn, which is C(26, 2) = 325. Then, you divide the number of favorable outcomes (1) by the total number of possible outcomes (325). The probability of drawing a specific pair of names is 1/325, which is approximately 0.0030769.

Q5: What is the difference between probability and odds?

A5: Probability and odds are related but distinct concepts. Probability is a measure of the likelihood of an event occurring, expressed as a value between 0 and 1. Odds, on the other hand, are a measure of the ratio of the probability of an event occurring to the probability of the event not occurring. For example, if the probability of drawing a specific pair of names from the hat is 1/325, the odds of drawing that pair are 1:324.

Q6: How do I calculate the probability of drawing a specific pair of names from the hat with replacement?

A6: To calculate the probability of drawing a specific pair of names from the hat with replacement, you need to calculate the total number of possible combinations of names that can be drawn, which is C(26, 2) = 325. Then, you divide the number of favorable outcomes (1) by the total number of possible outcomes (325). However, since the names are drawn with replacement, the total number of possible outcomes is actually 26 × 26 = 676. The probability of drawing a specific pair of names with replacement is 1/676, which is approximately 0.001474.

Q7: What is the probability of drawing two boys from the hat with replacement?

A7: To calculate the probability of drawing two boys from the hat with replacement, you need to calculate the number of ways to choose 2 boys from 10, which is C(10, 2) = 45. Then, you divide this number by the total number of possible outcomes, which is 26 × 26 = 676. The probability of drawing two boys with replacement is 45/676, which is approximately 0.066421.

Q8: How do I calculate the probability of drawing two girls from the hat with replacement?

A8: To calculate the probability of drawing two girls from the hat with replacement, you need to calculate the number of ways to choose 2 girls from 16, which is C(16, 2) = 120. Then, you divide this number by the total number of possible outcomes, which is 26 × 26 = 676. The probability of drawing two girls with replacement is 120/676, which is approximately 0.177684.

Conclusion

In this article, we have addressed some of the most frequently asked questions related to the probability of drawing specific pairs of names from a hat without replacement. We have provided step-by-step explanations and examples to help you understand the concepts and calculations involved. Whether you are a student, a teacher, or simply someone interested in probability, we hope this article has been helpful in clarifying your understanding of this important topic.

Final Thoughts

Probability is a fundamental concept in mathematics that has numerous applications in real-world scenarios. By understanding the basics of probability and how to calculate combinations and permutations, you can make informed decisions and predictions in a wide range of fields, from finance and economics to science and engineering. In the context of this problem, the probability of drawing specific pairs of names from a hat without replacement can be used to inform decisions about group project partners, and the probability of drawing two boys or two girls can be used to make predictions about the composition of the group.

References

  • [1] "Probability" by Khan Academy
  • [2] "Combinations and Permutations" by Math Is Fun
  • [3] "Probability and Statistics" by Coursera

Further Reading

  • [1] "Introduction to Probability" by Charles M. Grinstead and J. Laurie Snell
  • [2] "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole and Raymond H. Myers
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton