Using What You Learned In This Week's Lessons Answer The Fol

Using What You Learned In This Weeks Lessons Answer The Following Qu

Using what you learned in this week's lessons, answer the following questions: Give an example of an event in a random experiment and define its complement. What is the difference between discrete and continuous random variables? Include a real-world example in your answer that illustrates the difference between discrete and continuous random variables. Which of the following may be modeled by a binomial random variable? If the binomial random variable is not appropriate, give a reason why.

John is a 70% free-throw shooter. He decides to shoot 25 shots. We are interested in the number of successful shots. Assume the outcome on each shot is independent of all other shots. Margie is not very good at throwing darts. She is successful at hitting the bull’s eye about 30% of the time. We are interested in the number of throws it takes for her to hit the bull’s eye 10 times.

Paper For Above instruction

Introduction

Understanding the fundamental concepts of probability and statistics is crucial for analyzing and interpreting data in various real-world contexts. This paper explores key principles such as events and their complements, the distinction between discrete and continuous random variables, and the applicability of the binomial distribution to different scenarios. Through practical examples involving free-throw shooting and dart throwing, we illuminate these concepts and demonstrate their relevance.

Event and Complement in a Random Experiment

An event in a random experiment is a specific outcome or a set of outcomes that we are interested in observing. For instance, consider the experiment of flipping a coin. An event could be "getting heads," which occurs if the coin lands on heads after a flip. The complement of this event, denoted as "not getting heads" or "getting tails," encompasses all outcomes that are not part of the original event. In this case, the complement is the event "getting tails." The sum of the probabilities of an event and its complement always equals 1, reflecting the fact that these two outcomes encompass all possible outcomes of the experiment.

Discrete vs. Continuous Random Variables

Random variables are functions that assign numerical values to outcomes of a random experiment. They are classified as either discrete or continuous based on the nature of their possible values. Discrete random variables take on countable, often integer, values. For example, the number of successful free throws out of 25 attempts is a discrete variable because it can only take values like 0, 1, 2, ..., 25. Conversely, continuous random variables can assume any value within a given interval. An example of this is the height of students in a classroom, where measurements can be infinitely precise within a range, such as 150.2 cm or 175.8 cm. The key difference lies in their domain: discrete variables have countable outcomes, while continuous variables have uncountably many.

A real-world example contrasting these types involves the number of emails received per day (discrete) versus the exact amount of time (in minutes) it takes to commute to work (continuous). The former counts discrete instances; the latter measures a continuous quantity that can be measured with increasing precision.

Modeling with the Binomial Distribution

The binomial distribution is appropriate in scenarios where there are fixed numbers of independent trials, each trial has two possible outcomes (success or failure), and the probability of success remains constant across trials. In evaluating whether a particular scenario fits this model, we examine whether these conditions are met.

Scenario 1: John’s Free-Throw Shots

John's situation—trying 25 independent free throws with a success probability of 70%—fits the criteria for a binomial distribution. Each shot is a trial, with success defined as making the shot; the probability remains constant at 0.7; and the number of attempts is fixed at 25. Therefore, the number of successful shots John makes can be modeled by a binomial random variable, which allows calculation of probabilities such as the likelihood of at least 20 successful shots.

Scenario 2: Margie’s Dart Hitting

The problem with Margie’s dart throws—the number of throws needed to hit the bull’s eye 10 times—is not suitable for a binomial model. This scenario involves a negative binomial distribution, which models the number of trials needed to achieve a fixed number of successes, given constant probability per trial. Since we are counting the number of trials until reaching 10 successes, the binomial distribution, which assumes a fixed number of trials, does not apply here.

Conclusion

In summary, understanding events and their complements provides foundational insight into probability. Recognizing whether a random variable is discrete or continuous helps in selecting appropriate statistical models and in interpreting data accurately. The binomial distribution effectively models scenarios with fixed trials, constant success probability, and binary outcomes, as demonstrated with John’s free throws, but is unsuitable for cases like Margie’s dart throws where the number of trials varies until achieving a set number of successes. Mastery of these concepts enhances analytical capabilities across different fields, from sports analytics to quality control.

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