Probability Problem Selecting Group Project Partners

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Understanding probability is crucial in various aspects of life, from making informed decisions to solving complex problems. In mathematics, probability helps us quantify the likelihood of an event occurring. In this comprehensive guide, we will delve into a probability problem involving the selection of partners for a group project. We will explore the steps to solve the problem and discuss the underlying concepts. This discussion will involve understanding how to calculate probabilities in scenarios with and without replacement, which is a fundamental concept in probability theory. The problem presented involves a classroom scenario where Eduardo needs to select two partners from a hat containing the names of his classmates. By dissecting this problem, we aim to provide a clear understanding of how to approach and solve similar probability questions. This guide is designed to be accessible to students and anyone interested in enhancing their understanding of probability.

Problem Statement

In Eduardo's class, there are 26 other students. Their names are written on slips of paper and placed in a hat. Among these students, 10 are boys. To form groups for a project, Eduardo needs to randomly select two names from the hat without replacing the first name. This scenario introduces the concept of dependent events, where the outcome of the first event affects the outcome of the second event. The key to solving this problem lies in understanding how to calculate probabilities when selections are made without replacement. We need to consider how the total number of students and the number of boys change after the first name is drawn. The subsequent sections will explore the different probabilities that can be calculated based on this scenario, providing a step-by-step approach to solving each. This problem not only tests the understanding of probability but also the ability to apply it in a practical, real-world context. Understanding the nuances of such problems is crucial for mastering probability and statistics.

Understanding the Basics of Probability

To tackle this problem effectively, a solid understanding of basic probability principles is essential. Probability, in its simplest form, is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The basic formula for probability is:

Probability (Event) = (Number of favorable outcomes) / (Total number of possible outcomes)

In the context of our problem, a "favorable outcome" could be selecting a boy's name, while the "total number of possible outcomes" is the total number of names in the hat at any given selection. It is important to distinguish between independent and dependent events. Independent events are those where the outcome of one event does not affect the outcome of another (e.g., flipping a coin multiple times). Dependent events, on the other hand, are events where the outcome of one event influences the outcome of subsequent events. Our problem falls into the latter category since Eduardo is selecting names without replacement, meaning the pool of names changes after each selection. This change affects the probability of the next selection. Understanding these fundamental concepts is crucial for accurately calculating the probabilities in the given scenario. The next sections will build upon these basics to address specific probability questions related to Eduardo's group selection process.

Key Questions to Address

Based on the problem statement, several probability questions can be formulated. These questions will help us explore different aspects of probability calculations in the given scenario. Here are some key questions we will address:

  1. What is the probability that Eduardo selects two boys?
  2. What is the probability that Eduardo selects two girls?
  3. What is the probability that Eduardo selects one boy and one girl?

Each of these questions requires a slightly different approach in calculating the probability. The first question involves calculating the probability of two dependent events both being favorable (selecting a boy). The second question mirrors this but focuses on selecting girls. The third question is a bit more complex as it involves two possible scenarios: selecting a boy first and then a girl, or selecting a girl first and then a boy. We will need to consider both scenarios and add their probabilities to get the final answer. By addressing these questions, we will gain a comprehensive understanding of how to apply probability principles to solve real-world problems. The following sections will provide detailed solutions to each of these questions, along with explanations of the steps involved.

Solving for the Probability of Selecting Two Boys

To calculate the probability of Eduardo selecting two boys, we need to consider the probabilities of each selection sequentially. Let's break it down step by step:

  1. Probability of selecting a boy on the first draw:

    • There are 10 boys in the class.
    • There are 26 total students.
    • The probability of selecting a boy on the first draw is 10/26.

  2. Probability of selecting a boy on the second draw, given that a boy was selected on the first draw:

    • After selecting one boy, there are now 9 boys left.
    • There are also 25 total students remaining in the hat.
    • The probability of selecting a boy on the second draw is 9/25.

To find the probability of both events happening, we multiply the probabilities of each event:

Probability (Two Boys) = (10/26) * (9/25)

Simplifying the fractions:

Probability (Two Boys) = (5/13) * (9/25)

Probability (Two Boys) = 45/325

Further simplification gives us:

Probability (Two Boys) = 9/65

Therefore, the probability that Eduardo selects two boys for the group project is 9/65. This calculation demonstrates the application of conditional probability, where the outcome of the first event affects the probability of the second event. The key is to adjust the numbers to reflect the change in the pool of candidates after the first selection.

Calculating the Probability of Selecting Two Girls

Now, let's determine the probability that Eduardo selects two girls for the group project. Similar to the previous calculation, we will break this down into steps:

  1. Probability of selecting a girl on the first draw:

    • There are 26 total students, and 10 are boys, so there are 26 - 10 = 16 girls.
    • The probability of selecting a girl on the first draw is 16/26.

  2. Probability of selecting a girl on the second draw, given that a girl was selected on the first draw:

    • After selecting one girl, there are now 15 girls left.
    • There are also 25 total students remaining in the hat.
    • The probability of selecting a girl on the second draw is 15/25.

To find the probability of both events occurring, we multiply the individual probabilities:

Probability (Two Girls) = (16/26) * (15/25)

Simplifying the fractions:

Probability (Two Girls) = (8/13) * (3/5)

Probability (Two Girls) = 24/65

Thus, the probability that Eduardo selects two girls for the group project is 24/65. This calculation, like the previous one, highlights the importance of adjusting probabilities based on prior events in scenarios without replacement. The number of girls and the total number of students decrease after the first girl is selected, which directly impacts the probability of selecting another girl.

Determining the Probability of Selecting One Boy and One Girl

Calculating the probability of Eduardo selecting one boy and one girl is a bit more intricate because there are two possible scenarios:

  1. Selecting a boy first, then a girl.
  2. Selecting a girl first, then a boy.

We need to calculate the probability of each scenario and then add them together to get the total probability. Let's break it down:

Scenario 1: Boy then Girl

  • Probability of selecting a boy first: 10/26
  • Probability of selecting a girl second (given a boy was selected first): 16/25
  • Probability of Scenario 1: (10/26) * (16/25) = (5/13) * (16/25) = 80/325

Scenario 2: Girl then Boy

  • Probability of selecting a girl first: 16/26
  • Probability of selecting a boy second (given a girl was selected first): 10/25
  • Probability of Scenario 2: (16/26) * (10/25) = (8/13) * (2/5) = 80/325

Now, we add the probabilities of the two scenarios:

Probability (One Boy and One Girl) = Probability (Scenario 1) + Probability (Scenario 2)

Probability (One Boy and One Girl) = 80/325 + 80/325

Probability (One Boy and One Girl) = 160/325

Simplifying the fraction:

Probability (One Boy and One Girl) = 32/65

Therefore, the probability that Eduardo selects one boy and one girl for the group project is 32/65. This calculation demonstrates the importance of considering all possible scenarios when calculating probabilities for events that can occur in multiple ways. By breaking down the problem into manageable scenarios and then combining the results, we can accurately determine the overall probability.

Conclusion

In this comprehensive guide, we explored a probability problem involving the selection of partners for a group project. We addressed three key questions:

  1. The probability of selecting two boys (9/65).
  2. The probability of selecting two girls (24/65).
  3. The probability of selecting one boy and one girl (32/65).

Through these calculations, we reinforced the understanding of basic probability principles, including the crucial distinction between independent and dependent events. We also highlighted the importance of considering all possible scenarios when calculating probabilities for events that can occur in multiple ways. This guide provides a step-by-step approach to solving probability problems, emphasizing the need to adjust probabilities based on prior events in scenarios without replacement. By mastering these concepts, students and anyone interested in probability can confidently tackle similar problems and apply these principles in various real-world situations. Probability is a fundamental tool in decision-making and problem-solving, and a solid understanding of its principles is invaluable in many fields.

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