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Using R to Compute Probabilities of Matching Birthdays

Using R to Compute Probabilities of Matching Birthdays. Bruce E. Trumbo* Eric A. Suess Clayton W. Schupp †. California State University, Hayward JSM, August 2004 † Presenting at JSM * btrumbo@csuhayward.edu. Statement of Birthday Problem. 25 randomly chosen people in a room

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Using R to Compute Probabilities of Matching Birthdays

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  1. Using R to Compute Probabilities of Matching Birthdays

  2. Bruce E. Trumbo*Eric A. SuessClayton W. Schupp† California State University, Hayward JSM, August 2004 † Presenting at JSM * btrumbo@csuhayward.edu

  3. Statement of Birthday Problem • 25 randomly chosen people in a room • What is the probability two or more of them have the same birthday? • Model simplifications for a simple combinatorial solution: • Ignore leap years • Assume all 365 days equally likely • What error from assumptions?

  4. Let X = number of matches Seek P{X≥ 1} = 1 – P{X = 0} Seek P{X≥ 1} = 1 – 0.4313 = 0.5687 In R: prod(1 - (0:24)/365) Returns 0.4313003 Combinatorial Solution

  5. For n People in the Room • As n increases, clearly P{X≥ 1} increases. • If n = 366 (still ignoring leap years), we are sure to get at least one duplication. • But we will see P{X≥ 1} becomes very close to 1 for much smaller values of n. • Easy to show using a loop in R.

  6. R Script for Graph of P{Match} p <- numeric(50) for (n in 1:50){ q <- 1 - (0:(n - 1))/365 p[n] <- 1 - prod(q) } plot(p) # Makes Figure 1 p # Makes prinout below > p [1] 0.000000000 0.002739726 ... [21] 0.443688335 0.475695308 [23] 0.507297234 0.538344258 [25] 0.568699704 0.598240820 ... [49] 0.965779609 0.970373580

  7. Simulating the Birthday Process • Use R to “survey” from m = 10000 rooms, each with n = 25 randomly chosen people. • Two convenient functions in R: b <- sample(1:365, n, repl=T) samples 25 objects from among {1, 2, …, 365} with replacement, length(unique(b)) counts the number of unique objects.

  8. R Code Simulating the Birthday Process n <- 25; m <- 10000; x <- numeric(m) for (i in 1:m) { b <- sample(1:365, n, repl=T) x[i] <- n - length(unique(b)) } mean(x); mean(x==0) cut <- (0:(max(x) + 1)) - 0.5 hist(x, breaks=cut, freq=F, col=8) > mean(x) [1] 0.8081 > mean(x==0) [1] 0.4338

  9. Comments on Simulation Results > mean(x) [1] 0.8081 The proportion of rooms with no matches is very close to P{X = 0} = 0.4313. > mean(x==0) [1] 0.4338 The average number of matches per room simulates E(X), which is not easily obtained by combinatorial methods.

  10. How Serious Are Simplifying Assumptions? • We hope the 25 people in our room are randomly chosen. (For example, not a convention of twins, or Sagittarians.) • We know there are February 29th birthdays and that other birthdays are not uniformly distributed. We hope this does not matter much.

  11. Simulation Easy to Generalize Define a vector of weights: w <- c(rep(4,200), rep(5,165), 1) More extreme variation than actual. Sample according to these weights: sample(1:366, n, repl=T, prob=w) Slight modification of the R code simulates the nonuniform problem.

  12. R Code Simulating the Birthday Process With Weighted Probabilities n <- 25; m <- 20000; x <- numeric(m) w <- c(rep(4,200), rep(5,165), 1) for (i in 1:m) { b <- sample(1:366, n, repl=T, prob=w) x[i] <- n - length(unique(b)) } mean(x); mean(x==0) Uniform Simulation > mean(x) [1] 0.819 [1] 0.8081 > mean(x==0) [1] 0.42215 [1] 0.4338

  13. Breakdown in Birthday Problem • We have seen that mild departures from the uniform do not change substantially. • What about for extreme departures from the uniform? • Assume that half the birthdays are distributed over a period of k weeks (k = 1, . . . , 26). • Remainder are uniformly distributed over the rest of the year.

  14. Simulated Departures From Uniform(Red Line Indicates Uniform Distribution)

  15. Concluding Remarks • Simulation allows us to assess assumptions. Here the result was that realistic departures from the uniform are relatively harmless. • Simulating a case with a known answer builds confidence in the simulation model to be used for generalizing results. • Using R, simulations are easy enough to use in practical applications and in the classroom.

  16. References Peter Dalgaard: Introductory Statistics with R, 2002, Springer. William Feller: An Introduction to Probability Theory and Its Applications, Vol. 1, 1950 (3rd ed., 1968), Wiley. Thomas. S. Nunnikhoven: A birthday problem solution for nonuniform frequencies, The American Statistician, 46, 270–274, 1992. Jim Pitman and Michael Camarri: Limit Distributions and Random Trees Derived From the Birthday Problem With Unequal Probabilities,Electron. J. Probab. 5 Paper 2, 1-18, 2000

  17. Web Resources R software and manuals are available from www.r-project.org Birth frequencies are based on raw monthly totals are available at www.cdc.gov/nchs/products/pubd/vsus/vsus.html Auxiliary instructional materials for this article are posted at www.sci.csuhayward.edu/~btrumbo/bdmatch

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