Analysis of Bernoulli

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Florida Institute of Technology *

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3161

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Statistics

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Feb 20, 2024

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Analysis of Bernoulli's Principle Through Coin Flipping Abstract This report presents an investigation into Bernoulli's principle by conducting a series of coin flips. The objective was to analyze the probability distribution of outcomes and to verify the law of large numbers. The experiment involved flipping a fair coin 1000 times, recording the outcomes, and comparing the empirical probability with the theoretical probability. Introduction Bernoulli's principle, named after Swiss mathematician Jacob Bernoulli, is a fundamental theorem in probability theory. It describes the outcome of a random experiment that can result in only one of two possible outcomes, termed success or failure. This principle is crucial in understanding binary variables and binary outcomes in experiments. The coin flip experiment is a practical demonstration of Bernoulli's principle, where the two outcomes are heads or tails, each with a probability of 0.5 in a fair coin. Theory A Bernoulli trial is a random experiment where there are only two possible outcomes: success (S) or failure (F). The probability of success is denoted by � p and that of failure by =1− � � q =1− p . The Bernoulli distribution is a discrete probability distribution, which is used to model the number of successes in a single Bernoulli trial. The expected value (mean) of a Bernoulli random variable is � p , and the variance is �� pq . Methodology Materials  A fair coin

Procedure 1. The coin was flipped vertically with a consistent force by the same person to maintain uniformity in the flipping process. 2. Each outcome (head or tail) was recorded immediately after the flip. 3. This process was repeated for a total of 1000 flips. 4. The number of heads and tails were counted and used to calculate the empirical probability of each outcome. Results The results section would typically include a table summarizing the number of heads and tails, followed by a calculation of the empirical probabilities based on the observed frequencies. Graphs such as histograms or pie charts could be used to visually represent the distribution of outcomes. Example Table: Outcome Frequency Empirical Probability Heads 510 0.51 Tails 490 0.49 Discussion The results indicate that the empirical probabilities for heads and tails are close to the theoretical probability of 0.5. This observation supports the law of large numbers, which states that as the number of trials increases, the empirical probability of an event will converge to the theoretical probability. Minor deviations from the expected probability can be attributed to the randomness inherent in the experiment and the finite number of trials. Conclusion The coin flip experiment effectively demonstrates Bernoulli's principle and the law of large numbers. The observed empirical probabilities for heads and tails were found to be in close agreement with the theoretical probabilities, validating the concept of Bernoulli trials in a practical scenario. Future experiments could involve

increasing the number of trials or exploring other Bernoulli processes to further investigate the reliability of statistical predictions. References  Bernoulli, J. (1713). Ars Conjectandi .  Ross, S.M. (2010). A First Course in Probability . Pearson. This simplified lab report covers the essentials of conducting and reporting on a Bernoulli experiment. Depending on the specific requirements and guidelines provided, additional details and sections such as an in-depth literature review, more comprehensive methodology, detailed data analysis, and extended discussions on the implications of the findings could be incorporated into a full- length report.

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Analysis of Bernoulli

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