### 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

Thanks to Cory Doctorow of BoingBoing for his favorable review of Secret Codes!

Other Books

- Magnetocapacitive Tunnel Junctions - August 21, 2017

- Tardigrades - August 14, 2017

- Roman Concrete - August 7, 2017

- Solar Spicules - July 31, 2017

- Schroeder Diffuser - July 24, 2017

- Rough Microparticles - July 17, 2017

- Robot Musicians - July 10, 2017

- Walter Noll (1925-2017) - July 6, 2017

- cosmogony - July 3, 2017

- Crystal Prototypes - June 29, 2017

- Voice Synthesis - June 26, 2017

- Refining Germanium - June 22, 2017

- Granular Capillarity - June 19, 2017

- Kirchhoff–Plateau Problem - June 15, 2017

- Self-Assembly - June 12, 2017

- Physics, Math, and Sociology - June 8, 2017

- Graphene from Ethylene - June 5, 2017

- Crystal Alignment Forces - June 1, 2017

- Martian Brickwork - May 29, 2017

- Carbon Nanotube Textile - May 25, 2017

- The Scent of Books - May 22, 2017

- Patterns from Randomness - May 18, 2017

- Terpene - May 15, 2017

- The Physics of Inequality - May 11, 2017

- Asteroid 2015 BZ509 - May 8, 2017

- Fuzzy Fibers - May 4, 2017

- The Sofa Problem - May 1, 2017

- The Wisdom of Composite Crowds - April 27, 2017

- J. Robert Oppenheimer and Black Holes - April 24, 2017

- Modeling Leaf Mass - April 20, 2017

- Easter, Chicks and Eggs - April 13, 2017

- You, Robot - April 10, 2017

- Collisions - April 6, 2017

- Eugene Garfield (1925-2017) - April 3, 2017

- Old Fossils - March 30, 2017

- Levitation - March 27, 2017

- Soybean Graphene - March 23, 2017

- Income Inequality and Geometrical Frustration - March 20, 2017

- Wireless Power - March 16, 2017

- Trilobite Sex - March 13, 2017

- Freezing, Outside-In - March 9, 2017

- Ammonia Synthesis - March 6, 2017

- High Altitude Radiation - March 2, 2017

- C.N. Yang - February 27, 2017

- VOC Detection with Nanocrystals - February 23, 2017

- Molecular Fountains - February 20, 2017

- Jet Lag - February 16, 2017

- Highly Flexible Conductors - February 13, 2017

- Graphene Friction - February 9, 2017

- Dynamic Range - February 6, 2017

- Robert Boyle's To-Do List for Science - February 2, 2017

- Nanowire Ink - January 30, 2017

- Random Triangles - January 26, 2017

- Torricelli's law - January 23, 2017

- Magnetic Memory - January 19, 2017

- Graphene Putty - January 16, 2017

- Seahorse Genome - January 12, 2017

- Infinite c - January 9, 2017

- 150 Years of Transatlantic Telegraphy - January 5, 2017

- Cold Work on the Nanoscale - January 2, 2017

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**