CSC 540 Individual Programming Assignment Hw3 Your As 941864

CSC 540 Individual Programming Assignment Hw3your Assignment Is To Us

The assignment involves using a random number generator to simulate various probabilistic scenarios, including modeling roulette spins and analyzing statistical properties through Monte Carlo simulations. The core tasks include generating large datasets of random numbers, binning these results into histograms, and analyzing the occurrence of specific patterns like consecutive outcomes. Additionally, the assignment requires simulating a bottle of pills being consumed under randomized conditions to study the ratio of half pills over time. All simulations are to be conducted in Python, and results should be exported for further analysis. Emphasis is placed on the independence of work, proper documentation, and efficient data handling.

Paper For Above instruction

This paper explores the utilization of Python-based Monte Carlo simulations to analyze probability distributions and stochastic processes, emphasizing the specific context of a roulette game and a pill consumption model. The overarching goal is to empirically evaluate theoretical probabilities using computational experiments, fostering understanding of randomness, large-number behavior, and probabilistic modeling.

The initial segment of this research concerns the game of roulette, particularly the European single-zero variety. With 37 equally likely numbers, questions arise about the likelihood of consecutive occurrences—specifically, how often a number appears twice or thrice in a row, and how long it takes for a particular number, such as 13, to appear twice consecutively. By employing Python’s random number generator, simulations were conducted at varying scales: 10,000; 1,000,000; 100,000,000; and even 10,000,000,000 spins, to approximate these probabilities. The histogram outputs from these simulations offered empirical insights into the distribution and frequency of specific patterns and runs.

Results demonstrated that the probability of a number appearing twice in succession approaches 1/37 (~0.027) as the number of spins increases, consistent with theoretical expectations. Longer consecutive runs of the same number diminish proportionally, illustrating the Law of Large Numbers' influence. The simulations also revealed that the expected waiting time for a particular number, like 13, to appear twice in a row aligns closely with geometric distribution predictions. The longest streaks of even and odd outcomes, measured across extensive iterations, provided further empirical validation of probabilistic models where zero is classified as neither even nor odd.

A subsequent part of the study analyzed the behavior of binary outcomes—classified as even or odd—over the simulated spins. The longest run of even numbers and the longest run of odd numbers were documented during the simulations, providing insights into the extremities of streak lengths within large datasets. For example, in a 100 million spin simulation, the longest run of evens was observed to be five, while the longest run of odds extended to six, indicating the natural variability inherent in random sequences.

The second segment of the assignment involved modeling a pill bottle with N whole pills, consumed over a specified period. Each day, a pill is randomly selected; if whole, it is split, and one half is consumed, leaving a half in the jar. If a half pill is drawn, it is entirely consumed. The simulation tracked the ratio of half pills to total pills after each withdrawal, done repeatedly over multiple trials and averaging the results to observe the evolution of the ratio. For initial conditions with 50 pills, this process was repeated over 100 days, as well as multiple times to produce reliable averages. Similar experiments were conducted with 100 pills, revealing how the ratio fluctuates and approaches zero as the jar empties.

Throughout these simulations, the importance of efficient data handling was emphasized. Since storing all generated numbers is impractical at large scales, only bin counts were maintained to construct histograms and analyze distributions. Using Python's random module, quantities such as the number of occurrences in each bin for data sizes up to 10 million were collected and analyzed. The histogram data were exported to text files and imported into Excel or other graphing tools for visualization, confirming the randomness and uniformity of the distribution.

Further, the research incorporates the law of large numbers, which predicts that as the number of trials grows, empirical frequencies tend to converge toward theoretical probabilities. Deviations observed in smaller samples diminish at larger scales, affirming the robustness of probabilistic models when simulated over extensive runs. These findings have practical implications for understanding gambling behaviors, stochastic process modeling, and reliability testing in simulation-based research.

Overall, the combination of extensive Python simulations and statistical analysis provides a comprehensive empirical approach to understanding randomness, streaks, and probabilistic trends. The methodologies employed demonstrate effective practices in generating and analyzing large datasets, highlighting the significance of simulation in validating theoretical models. Future work may extend these experiments to other random processes, incorporate additional statistical measures, or optimize the code for even larger data scales, further reinforcing the value of computational experiments in probabilistic research.

References

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