# How-To Geek

#### GEEK TRIVIA

A Birthday |
Blood Type |

Great-Great-Great Cousins |
Ear Wax Type |

**Answer: A Birthday**

In the field of probability theory, there is a problem known as the Birthday Paradox concerning the probability that in a selection of *N* randomly chosen people, some of them will share a birthday. This probability reaches 100% once you reach a sample size of 367 (to account for the 366 potential days, including Feb. 29, +1).

What’s fascinating, however is how quickly the probability of climbs. In a group of 23 people, around the size of your average primary or secondary classroom, the chances of two people sharing a birthday has already climbed to 50%. To get to 99%, you’d just need to gather two classrooms together. In a group of 57 people there is a 99% chance there is a common birthday. The change between 57 and 367 people is a hundredth of a percent per person added to the pool.

While this might seem like something of a mathematical parlour trick, the math behind the Birthday Paradox has actually been successfully employed as a well known cryptographic attack, the Birthday Attack, which uses probabilistic modeling to reduce the complexity of cracking encryption hash functions.

I've heard the number of people can be as few as thirteen; in a group of thirteen people, two will share a birthday. Winston Churchill, the Prime Minister and not the American Author, at a dinner party of twelve people tried this trick--and alas it failed, until a serving girl mentioned that she made the thirteenth person in the group and she shared her birthday with another person in the group. I'm working from memory of what I read almost 50 years ago in Walter Thompson's book on being Churchill's bodyguard. Or was it Lord Moran's book on being Churchill's doctor.? Ah yes, I remember it well.

In the Pizza Hut one day I had a discussion with a knowledgeable friend who knew the paradox that 23 folks were needed for a 50-50 chance. I tried to convince him it was only eight. After a lively discussion I bet him that between our table of four and the table next to us there would be a match. Knowing the odds were greatly against me he readily took the bet. However, he could not take into account the fact that I already knew someone at the next table had the same birthday as mine. Bet won.

My first question is, has the availability of computers made the calculation a bit more accurate. In 1954, I was in a Mathematics of Finance class at the University of Kentucky. The professor told us it took 24 persons. So I wonder if rounding to reduce the calculation effort could have produced that small difference.

In our families, my wife and I have what I think is an unusually large number of persons with common birthdays. I have two brothers (10 years a part) with the same birthday. My wife has a sister (nine years younger) that shares her birthday. Two of our children (three years a part) share the same birthday. The daughters with the shared birthday also share their birthday with the older ones brother-in-law. Her mother shares a birthday with her mother-in-law. One of my brothers has two children (not twins) who share the same birthday.

While 13 random people in a room might have two people share the same birth date, the probability of it isn't 50% in any random 13-member group. That's all it's saying.

Not so. This implies that the rate of change between 57 and 367 people is linear. It is not, as the graph is not a straight line.

My classroom has 47 people. Wut?

Then the odds should be much better.

They are. Two pairs of people share a birthday. (one pair are twins, though...)