The Birthday Problem

One of the most fun statistical results, is The Birthday problem, and it is a favorite for Intro to Probability courses.

Imagine you're at a party with 23 other guests. If you randomly pull aside one of them, the odds he or she will share your birthday are 1 in 365. But the probability that anyone at the party shares a birthday is far higher: about 1 in 2. The intuition is that the probability someone shares a birthday, is not 1/365 times the number of people present, but rather, the probability two people do not share a birthday (364/365) raised to the power of the number of pairs of people. For example, consider the number of combinations of people which is about (N^2)/2 for a sample of N people. So the probability that no one shares a birthday, with N people, is about (364/365)^[(N^2)/2]=49%ish.

Thus it was fun to read lawyers getting all excited about a new piece of research refuting DNA statistics by noting that some people share 7 o 13 gene loci at about 1 million times the frequency presented, a large error. But the error was the researcher. The odds are for a particular set of 13 loci sharing 9 chromosomes. But state crime lab analyst Kathryn Troyer looked at all the combinations of men in a group of 63,000, giving her about 2 billion pairs, and thought she discovered a massive rejection of the hypothesis. She also appears to have looked at many different sets of 13 loci, and ignored the fact that some prisoners are related.

Steven Myers, a senior DNA analyst at the California Department of Justice, gave a presentation to the Assn. of California Crime Lab Analysts, titled "Don't Panic".

Lawyers. They're so cute when they try to do statistics.