### The Birthday Problem

August 4, 2010 Today is the birthday of one of my family members. To celebrate, I'm reviewing a bit of mathematics called the "Birthday Problem." I was introduced to the Birthday Problem when I was in high school. I attended a weekly mathematics seminar, called the Colgate Seminar, taught by a rotating corp of professors from nearby Colgate University. The Birthday problem is an example problem that's often used with high school students. The math isn't difficult, and the result is surprising. The problem is this - how many people do you need in a room, such that it's more likely than not that two of them have the same birthday? The important piece of the problem is not that any person has a particular birthday, or any person has the same birthday as one particular person; rather, that any two people will have the same birthday. To make things simple, disregard leap years, consider just 365 days in a year, and ignore twins. Surprisingly, there was a set of twins in my seminar session. The mathematics is quite simple. We use the principle that we can calculate the probability of independent events happening at the same time by multiplying their separate probabilities. We start with the first pair of people, N = i and N = (i + 1). The probability that person (i + 1) has a different birthday than person i is 364/365; that is, person (i + 1) must be born on any of the remaining 364 days that are not the birthday of person i. Bringing in another person (i + 2), and comparing him with persons i and (i + 1) gives us a probability of 363/365 that his birthday differs from the previous people. Continuing the calculationP = (364/365)(363/365)(362/365)...or, in compact notation where P(N) is the probability that in a group of N people, no two will have the same birthday. Of course, what we want is (1 - P(N)), the probability that two will have the same birthday. As you can see from the table, not that many people are needed to have just a 50:50 chance. It takes just 23 people to have a 50.7% probability that two will have the same birthday.

N | P(N) | 1 - P(N) |

5 | 0.97286 | 0.02714 |

10 | 0.88305 | 0.11695 |

15 | 0.74710 | 0.25290 |

20 | 0.58856 | 0.41144 |

25 | 0.43130 | 0.56870 |

30 | 0.29368 | 0.70632 |

35 | 0.18562 | 0.81438 |

40 | 0.10877 | 0.89123 |

45 | 0.05902 | 0.94098 |

50 | 0.02963 | 0.97037 |

^{N}codes, you'll get a collision not after 2

^{N}codes are generated, but rather after only 2

^{N/2}codes are generated. There is, in fact, a so-called birthday attack on hash functions; viz., generating multiple messages and finding a collision between an arbitrary pair of them is far easier than generating a document that has the same hash value as a particular document. For example, you may want to generate two nearly similar contracts that have the same hash value (a.k.a., digital signature), so you insert commas, extra spaces or blank lines, or use synonyms for words, until you get a collision. Then, one contract can be substituted for the other at a later time. I mentioned my high school interest in mathematics (see the figure). I did pursue a career in a mathematically intensive field; but I didn't particularly care for mathematics instruction, so I never considered becoming a mathematician. In later life, I've rediscovered some interesting mathematics, and I'm happy that I had enough math education for me to do some independent study.

### References:

*Permanent Link to this article*

Linked Keywords: Colgate Seminar; Colgate University; probability; cryptographic hash functions; hash collisions; birthday attack

RSS Feed

### Google Search

Latest Books by Dev Gualtieri

**Previews Availableat Tikalon Press**

*STEM-themed novel for middle-school students*

*Mathematics-themed novel for middle-school students*

*Complete texts of LGM, Mother Wode, and The Alchemists of Mars*

Other Books

- Adhesives, December 4, 2023

- MOND and Planet Nine, November 27, 2023

- Earthquake Light, November 20, 2023

- Antimatter Gravitation, November 13, 2023

- Koch Snowflake, November 6, 2023

- Shared Dessert, October 30, 2023

- Quantum Uncertainty, October 23, 2023

- Sulfur Hexafluoride, October 16, 2023

- Cement Supercapacitors, October 9, 2023

- Fifth Force, October 2, 2023

- The Birthday Problem, September 25, 2023

- Iron as a Fuel, September 18, 2023

- Cosmic Asymmetry, September 11, 2023

- Work, September 4, 2023

- The Monty Hall Problem, August 28, 2023

- Memristors, August 21, 2023

- Aperiodic Tiling, August 14, 2023

- Fondant Physics, August 7, 2023

- The Gravitational Constant, July 31, 2023

- Length of a Day, July 24, 2023

- Random Walks, July 17, 2023

- Gravitational Lensing, July 10, 2023

- Tiny Bubbles, July 3, 2023

- Thales' Measure of the Sun, June 26, 2023

- Tetrataenite Magnets, June 19, 2023

- Utilitarian Music, June 12, 2023

- Medieval Volcanism, June 5, 2023

- Rare Earths from Bacteria, May 29, 2023

- Georges Lemaitre, May 22, 2023

- Reading Old Manuscripts, May 15, 2023

- The IQ Flynn Effect, May 8, 2023

- Cosmic Water, May 1, 2023

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**