Birthday_Problem

If we have K balls, and M boxes, and you start throwing balls randomly onto boxes, then the probability that a box contains more than one ball is given by:

\[ 1 - e^\frac{-K^2}{2M} \]

• If 50 people gather in a room, what is the probability that at least 3 have the same birthday?

mean(replicate(10000, 3 %in% table(sample(1:365,50,replace=T))))

## [1] 0.1233

• If 50 people gather in a room, what is the probability that at least 2 have the same birthday?

mean(replicate(10000, 2 %in% table(sample(1:365,50,replace=T))))

## [1] 0.9619

*How many people are needed in a room so that at least 3 of them have the same birthday with 95% probability. Let us write a generalized function for this purpose. The function accepts the required number of people and probability, and number of days in the year as inputs:

birthday_min_ppl <- function(n,p,d)
  {
  
  i <- 2
  
  repeat 
    {
    if(mean(replicate(1000, n %in% table(sample(1:d,i,replace=T)))) >= p) return(i)
    i <- (i+1)
    }
  
  }

#If 3 or more people to have the same birthday (with 50% prob.), how many people are needed in the room?
birthday_min_ppl(3,.5,365)

## [1] 90

#If 2 or more people to have the same birthday (with 95% prob.), how many people are needed in the room?
birthday_min_ppl(2,.95,365)

## [1] 46

So we need approximately 90 people if 3 people to have the same birthday with 50% probability. We need 50 people if two or more people to have the same Birthday with 95% probability