Probability paradoxes behind Benford's Law and a Lottery Paradox

• Main Author: Wong, Chin Hao
• Other Authors: Tay Kian Boon
• Format: Final Year Project
• Language: English
• Published: Nanyang Technological University 2022
• Subjects: Engineering::Computer science and engineering
• Online Access: https://hdl.handle.net/10356/156605
• Institution: Nanyang Technological University

Probability Theory is a study of the likelihood of an event to occur. However, there are some probabilities where people might find it hard to believe as it conflicts with their understanding. These probabilities are called Paradoxes in Probability. There are many well-known paradoxes in probability theory, such as the Birthday Paradox, Monty Hall Problem, etc. This report will study two paradoxes in Probability Theory, mainly Benford’s Law and Lottery Paradox. Benford’s Law is an observation that states that the first digit of many numbers in naturally occurring real-world data does not follow a uniform distribution but instead follows a logarithmic distribution. In Benford’s Law, we will first briefly describe how this paradox was discovered first by Newcomb and then popularised by Benford. Secondly, we will prove why this phenomenon happens based on published papers by various researchers. Thirdly, we will provide some popular goodness-of-fit tests that people use to test for the compliance of data to Benford’s Law and some applications of how Benford’s Law can be used in the real world. Lastly, we will collect conduct some case studies where we will gather data from real-world to prove and support the existence of such phenomenon. Next, we will talk about the Lottery Paradox. Lottery Paradox is a phenomenon that states that in the lottery, consecutive numbers appear 50% of the time. In Lottery Paradox, we will first prove why this phenomenon happens by explaining the concept of combinatorial and stars and bars. Next, we will test this paradox by creating a small trial using a small Python program to randomise 800 lottery numbers. Our test finds that consecutive numbers occur around 50% of the time. Lastly, we will collect some real-world lottery data to support our claim that such paradox does happen in the real world.