# birthday: Probability of coincidences

Description Usage Arguments Details Value References Examples

### 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

 1 2 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.

### Value

 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.

### Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 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)

Questions? Problems? Suggestions? or email at ian@mutexlabs.com.

Please suggest features or report bugs with the GitHub issue tracker.

All documentation is copyright its authors; we didn't write any of that.