CSC 540 Individual Programming Assignment Hw3 Your Pr 433702

CSC 540 Individual Programming Assignment Hw3your Program Should Use

Explore the behavior of a Pascal random number generator by simulating random number throws within specified ranges and binning results into histograms. Perform multiple simulation runs with increasing sample sizes, produce histogram data, and analyze the distribution. Additionally, simulate a simplified model of soccer match outcomes based on goals scored derived from random algorithms, observing how win percentages vary with different parameters. Furthermore, model pill consumption over time with random selection, track the ratio of half pills, and analyze its progression. Finally, simulate roulette wheel spins to empirically determine probabilities of consecutive number appearances and streaks for specific numbers, analyzing data from varying total spins. Use the Free Pascal compiler, maintain appropriate data types such as qword for large numbers, document your code thoroughly with comments, and generate empirical data solely with your program for reporting and analysis.

Paper For Above instruction

In this comprehensive assignment, we explore various stochastic processes through simulation using the Free Pascal programming language. Each segment of the assignment is designed to model a different probabilistic scenario, allowing empirical analysis of outcomes that are otherwise difficult to predict analytically. The implementation emphasizes efficient data handling, proper use of Pascal's data types including qword for large numbers, and thorough documentation to ensure clarity and reproducibility of results.

Histogram Simulation of Random Number Generation:

The initial task involves generating pseudorandom integers within a fixed range (20001 to 380000) and binning the results into histograms. By executing this simulation multiple times across increasing sample sizes—1,000; 10,000; and 100,000 numbers—the goal is to observe the distribution of outcomes and verify the uniformity of the random number generator. The histogram data should be stored as counts per bin, avoiding the overhead of storing individual numbers. After each run, the histogram results are exported to a text file, which can be imported into Excel or another graphing tool to visualize the distribution as bar charts. Ensuring bins are evenly divided over the range guarantees each outcome has an equal probability.

2D Histogram with Larger Bin Counts:

Expanding on this, a two-dimensional histogram with 36 bins arranged in a 6x6 grid is constructed, covering a similar range. This spatial visualization enables analysis of any potential correlations or patterns in the distribution. Multiple runs with large sample sizes (like 100,000; 1,000,000; and 10,000,000) will observe how the empirical distribution approaches uniformity, illustrating the law of large numbers.

World Cup Soccer Simulation:

The second core component involves simulating soccer matches under simplified assumptions. Each team’s scoring per game is modeled by rounding a uniformly generated random number between 0 and twice their average goals (X1 and Y1 respectively). The formula purposefully introduces variability, despite its unrealistic nature, to examine how the frequency of wins and ties depend on the average goals and the number of matches (N). By fixing averages and varying N, we can observe how the percentage of wins converges towards expected probabilities as the number of simulations increases, consistent with the law of large numbers. Changing the average goals per game allows investigation into the impact on win, loss, and tie percentages, explored through plotting (X1 minus Y1) versus outcomes.

Pill Consumption Model:

The program then models daily pill consumption from a bottle initially containing N whole pills. Each day, a random pill is selected: if it’s whole, it’s split into halves, with one half used and the other returned; if a half pills is drawn, it’s consumed. This process continues until the bottle is exhausted. Tracking how the ratio of half pills to total pills evolves provides insight into the stochastic depletion process. Running multiple simulations with different initial counts (like 50 and 100 pills) and averaging results yields a more robust understanding of the typical ratio trajectory over time.

Roulette Wheel Simulation:

Finally, the simulation models the randomness of roulette draws, assuming each number (0-36) has equal probability. The program calculates ratios of consecutive hits, such as how often a number appears twice or thrice in a row, and averages these over a large number of spins (up to billions). It also measures how long it takes for a specific number (like 13) to appear twice in a row, and tracks the longest streaks of even or odd outcomes (treating zero as neutral). By doing multiple runs with increasing total spins, the empirical probabilities can be compared with theoretical expectations. The challenge lies in efficiently handling massive data counts with appropriate data types and minimizing I/O bottlenecks.

Throughout all these sections, the program emphasizes efficiency, correctness, and adherence to the Pascal language standards. Proper commenting, defining constants, and modularizing code via functions and procedures promote clarity and maintainability. This multifaceted empirical exploration demonstrates the power of simulation in understanding probabilistic phenomena and the importance of large sample sizes in achieving statistically significant results.

References

• Knuth, D. E. (1998). The Art of Computer Programming, Volume 2: Seminumerical Algorithms. Addison-Wesley.
• Mitzenmacher, M., & Upfal, E. (2005). Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press.
• Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). Numerical Recipes: The Art of Scientific Computing (3rd ed.). Cambridge University Press.
• Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1), 97-109.
• Grinstead, C. M., & Snell, J. L. (1997). Introduction to Probability. American Mathematical Society.
• Gordon, C. (2006). Simulation Modeling and Analysis. McGraw-Hill.
• Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Volume 1. Wiley.
• Norris, J. R. (1998). Markov Chains. Cambridge University Press.
• Grove, A. (2014). Python for Data Analysis. O'Reilly Media. (for concepts on data visualization and interpretation)
• Ross, S. M. (2006). Introduction to Probability Models. Academic Press.