Birthday Paradox

From SkepticWiki

The probability that two people in a group have the same birthday. It's an even bet with 23 people, and almost certain with 60.

[edit] Definition

The Birthday Paradox is a counterintuitive result in statistics and probability. It is commonly expressed as:

In a group of 60, people, it is almost certain (99.2%) that two have the same birthday.

This may seem counterintuitive, because we tend to erroneously interpret such a situation as comprising 60 “probability trials”. In fact, there are 1770 “trials” in this situation, representing all possible pairs of individuals.

[edit] Generalization

Given a set of n values chosen from a range of d values, the probability p that two are the same is approximately

[edit] Example

As an example, the probability of choosing the winning numbers in a 6/53 lotto is about 1 in 22 million. However in any group of 14,500 tickets, there is a 99% probability that two tickets have exactly the same numbers!

[edit] Fallacy

A fallacy occurs when one erroneously uses the probability that specific value is matched. In most cases, this would imply a much lower probability of

.

In the case of the matching lotto tickets, this would suggest that the probability of a match in 14,500 tickets is only 0.06%, a far cry from the actual 99% probability!

(Note: Another common fallacy is that the probability of an event occurring in n trials is n times the probability of a single trial. This would suggest a slightly higher p = n / d.)

This fallacy gives rise to apparent remarkable coincidences which, statistically speaking, are not really very remarkable. The phenomenon of “disasters come in threes” is a familiar example. (e.g.: Johnny Cash, June Carter Cash, and daughter Rosie Adams all died in 2003)

From the large universe of high-profile events and disasters, it is nearly certain for these types of apparent coincidences to occur. In these cases, fallacious reasoning may add intellectual justification to a superstitious interpretation of such events.

A non-discrete two-dimensional generalization of the birthday paradox is at the heart of the recognition of Ley Lines and other seemingly coincidental “alignments”.