When is your birthday? The math behind hash collisions
This article unpacks the surprisingly counter-intuitive math behind the 'Birthday Paradox,' revealing why the chances of shared birthdays are much higher than commonly perceived. It then delves into Richard von Mises's nuanced approach to 'occupancy probability,' which offers a more practical way to understand the likelihood of such events. Crucially for Hacker News readers, this mathematical framework directly underpins the concept of hash collisions and the mechanics of 'Birthday Attacks' in cybersecurity.
The Lowdown
The story begins by introducing the famous 'Birthday Paradox,' which states that in a group of just 23 people, there's a 50% chance that two individuals share the same birthday. It demystifies this counter-intuitive probability using straightforward school-level combinatorics, focusing on the inverse probability of no one sharing a birthday.
Key takeaways from the article include:
- Classical Birthday Paradox: Explains the calculation showing a 50% probability of a shared birthday in a group of 23 people, and how this is derived by calculating the probability of no shared birthdays and subtracting it from one.
- Limitations of Specific Event Probability: The article highlights the flaw in calculating the probability of a specific event, such as three people sharing a pre-chosen birthday, which yields a very low chance.
- Von Mises's Occupancy Probability: Introduces Richard von Mises's 1939 concept of 'occupancy probability,' which shifts the focus from a specific day to the expected number of days occupied by a certain number of birthdays within the entire set of possibilities.
- Re-evaluating Rarity: Applying von Mises's method, the article demonstrates that events like three people sharing a birthday in a group of 60 are far less rare than initially calculated, with an expected value of approximately 0.22 occurrences, meaning it happens about once in every 4-5 such groups.
- Real-World Applicability: The core mathematics directly translates to understanding hash collisions, where 'days' become hash table fields and 'people' become hashes.
- Cybersecurity Implications: The principles are critical to the 'Birthday Attack' in cybersecurity, a brute-force method that exploits the higher probability of finding any collision (not a specific one) to compromise systems much faster than other methods.
In conclusion, the article expertly guides the reader through complex probability concepts, moving from a common paradox to a sophisticated mathematical framework with significant practical implications for computer science and cybersecurity.