R: Probability of coincidences

birthday {base} | 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, Mosteller F. "Methods for studying coincidences" JASA 84:853-861

### Examples

     ## 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)
    ## 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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