The Birthday Problem

Probability Puzzles in R

Peter Chi

Assistant Professor of Statistics Villanova University

Problem overview

Setup

Room with n people in it
What is the probability that anyone shares the same birthday?

Assumptions

Ignore February 29th
Birthdays are uniformly distributed across the remaining 365 days
Each individual in the room is independent

Defining a counter

Simulating the probability of rolling a 12

counter <- 0

roll <- roll_dice(2)
roll

if(roll == 12){
  counter <- counter + 1
}

counter

roll_dice <- function(k){
  all_rolls <- sample(c(1,2,3,4,5,6), 
                      k, 
                      replace = TRUE)
  final_answer <- sum(all_rolls)
  return(final_answer)
}

Incrementing a counter in a loop

Simulating the probability of rolling a 12

counter <- 0

for(i in 1:10000){
  roll <- roll_dice(2)
  if(roll == 12){
    counter <- counter + 1
  }
}
p_twelve <- counter / 10000

print(p_twelve)

0.0278

The pbirthday function

pbirthday(10)

0.1169482

room_sizes <- c(1:10)
match_probs <- sapply(room_sizes, pbirthday)
print(match_probs)

0.000000000 0.002739726 0.008204166 0.016355912 0.027135574 0.040462484 0.056235703 0.074335292 
0.094623834 0.116948178

Plotting

plot(match_probs ~ room_size)

image-url

Let's solve it!

Probability Puzzles in R