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According to the famous law of large numbers, the empirical a very large number of times. probability approaches the probabilit

According to the famous law of large numbers, the empirical a very large number of times. probability approaches the probabilit

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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According to the famous law of large numbers, the empirical a very large number of times. probability approaches the probabilit
**Law of Large Numbers Explanation** According to the famous law of large numbers, the empirical probability approaches the theoretical probability when an experiment is repeated a very large number of times. This principle is crucial in statistics and probability theory, as it helps to understand that as the number of trials or observations increases, the average of the results obtained from these trials will get closer to the expected value. Consider an experiment where you are flipping a fair coin. The theoretical probability of obtaining heads is 0.5 or 50%. If you flip the coin only a few times, the empirical probability (observed frequency of heads) might vary significantly from 0.5. However, as you increase the number of flips to a very large number, the empirical probability will get closer and closer to the theoretical probability of 0.5. #### Graphs and Diagrams If we include a graph, it would typically show: - **X-axis (Horizontal)**: Number of Trials - **Y-axis (Vertical)**: Empirical Probability The graph would depict several points representing the empirical probability at different numbers of trials. Over many trials, the points would form a curve that gets closer to the line representing the theoretical probability. This graphical representation visually reinforces the concept that increasing the number of trials diminishes the difference between empirical and theoretical probabilities.
Transcribed Image Text:**Law of Large Numbers Explanation** According to the famous law of large numbers, the empirical probability approaches the theoretical probability when an experiment is repeated a very large number of times. This principle is crucial in statistics and probability theory, as it helps to understand that as the number of trials or observations increases, the average of the results obtained from these trials will get closer to the expected value. Consider an experiment where you are flipping a fair coin. The theoretical probability of obtaining heads is 0.5 or 50%. If you flip the coin only a few times, the empirical probability (observed frequency of heads) might vary significantly from 0.5. However, as you increase the number of flips to a very large number, the empirical probability will get closer and closer to the theoretical probability of 0.5. #### Graphs and Diagrams If we include a graph, it would typically show: - **X-axis (Horizontal)**: Number of Trials - **Y-axis (Vertical)**: Empirical Probability The graph would depict several points representing the empirical probability at different numbers of trials. Over many trials, the points would form a curve that gets closer to the line representing the theoretical probability. This graphical representation visually reinforces the concept that increasing the number of trials diminishes the difference between empirical and theoretical probabilities.
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