| % File src/library/stats/man/birthday.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 1995-2018 R Core Team |
| % Distributed under GPL 2 or later |
| |
| \name{birthday} |
| \alias{qbirthday} |
| \alias{pbirthday} |
| \title{Probability of coincidences} |
| \description{ |
| Computes answers to a generalised \emph{birthday paradox} problem. |
| \code{pbirthday} computes the probability of a coincidence and |
| \code{qbirthday} computes the smallest number of observations needed |
| to have at least a specified probability of coincidence. |
| } |
| \usage{ |
| qbirthday(prob = 0.5, classes = 365, coincident = 2) |
| pbirthday(n, classes = 365, coincident = 2) |
| } |
| \arguments{ |
| \item{classes}{How many distinct categories the people could fall into} |
| \item{prob}{The desired probability of coincidence} |
| \item{n}{The number of people} |
| \item{coincident}{The number of people to fall in the same category} |
| } |
| \value{ |
| \item{qbirthday}{ |
| Minimum number of people needed for a probability of at least |
| \code{prob} that \code{k} or more of them have the same one out of |
| \code{classes} equiprobable labels. |
| } |
| |
| \item{pbirthday}{Probability of the specified coincidence.} |
| } |
| \details{ |
| The birthday paradox is that a very small number of people, 23, |
| suffices to have a 50--50 chance that two or more of them have the same |
| birthday. This function generalises the calculation to probabilities |
| other than 0.5, numbers of coincident events other than 2, and numbers |
| of classes other than 365. |
| |
| The formula used is approximate for \code{coincident > 2}. The |
| approximation is very good for moderate values of \code{prob} but less |
| good for very small probabilities. |
| } |
| \references{ |
| Diaconis, P. and Mosteller F. (1989). |
| Methods for studying coincidences. |
| \emph{Journal of the American Statistical Association}, \bold{84}, |
| 853--861. |
| \doi{10.1080/01621459.1989.10478847}. |
| } |
| \examples{ |
| require(graphics) |
| |
| ## the standard version |
| qbirthday() # 23 |
| ## probability of > 2 people with the same birthday |
| pbirthday(23, coincident = 3) |
| |
| ## examples from Diaconis & Mosteller p. 858. |
| ## 'coincidence' is that husband, wife, daughter all born on the 16th |
| qbirthday(classes = 30, coincident = 3) # approximately 18 |
| qbirthday(coincident = 4) # exact value 187 |
| qbirthday(coincident = 10) # exact value 1181 |
| |
| ## same 4-digit PIN number |
| qbirthday(classes = 10^4) |
| |
| ## 0.9 probability of three or more coincident birthdays |
| qbirthday(coincident = 3, prob = 0.9) |
| |
| ## Chance of 4 or more coincident birthdays in 150 people |
| pbirthday(150, coincident = 4) |
| |
| ## 100 or more coincident birthdays in 1000 people: very rare |
| pbirthday(1000, coincident = 100) |
| } |
| \keyword{distribution} |