# birthday {stats}

Probability of coincidences
Package:
stats
Version:
R 3.0.2

### Description

Computes answers to a generalised birthday paradox problem. `pbirthday` computes the probability of a coincidence and `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

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 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 `coincident > 2`. The approximation is very good for moderate values of `prob` but less good for very small probabilities.

### Values

qbirthday
Minimum number of people needed for a probability of at least `prob` that `k` or more 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.

### Note

Prior to R 2.14.0 the approximate formula was used even for `coincident = 2`.

### 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)```

Documentation reproduced from R 3.0.2. License: GPL-2.