birthday {stats}

R Documentation

Probability of coincidences

Description

Computes approximate answers to a generalised birthday paradox problem. pbirthday computes the probability of a coincidence and qbirthday computes the number of observations needed to have a specified probability of coincidence.

Usage

qbirthday(prob = 0.5, classes = 365, coincident = 2)
pbirthday(n, classes = 365, coincident = 2)

Arguments

classes	How many distinct categories the people could fall into
prob	The desired probability of coincidence
n	The number of people
coincident	The number of people to fall in the same category

Details

The birthday paradox is that a very small number of people, 23, suffices to have a 50-50 chance that two 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.

This formula is approximate, as the example below shows. For coincident=2 the exact computation is straightforward and may be preferable.

Value

qbirthday	Number of people needed for a probability prob that k of them have the same one out of classes equiprobable labels.
pbirthday	Probability of the specified coincidence

References

Diaconis, P. and Mosteller F. (1989) Methods for studying coincidences. J. American Statistical Association, 84, 853-861.

Examples

require(graphics)

 ## the standard version
qbirthday()
 ## same 4-digit PIN number
qbirthday(classes=10^4)
 ## 0.9 probability of three coincident birthdays
qbirthday(coincident=3, prob=0.9)
## Chance of 4 coincident birthdays in 150 people
pbirthday(150,coincident=4)
## 100 coincident birthdays in 1000 people: *very* rare:
pbirthday(1000, coincident=100)

## Accuracy compared to exact calculation
x1<-  sapply(10:100, pbirthday)
x2<- 1-sapply(10:100, function(n)prod((365:(365-n+1))/rep(365,n)))
par(mfrow=c(2,2))
plot(x1, x2, xlab="approximate", ylab="exact")
abline(0,1)
plot(x1, x1-x2, xlab="approximate", ylab="error")
abline(h=0)
plot(x1, x2, log="xy", xlab="approximate", ylab="exact")
abline(0,1)
plot(1-x1, 1-x2, log="xy", xlab="approximate", ylab="exact")
abline(0,1)

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