## birthday paradox, one by one

An easy riddle, riding the birthday paradox. Namely how many people on average have to sequentially enter a room for a double birthday to occur?

$\sum_{n=2}^{366}n\prod_{m=1}^{n-2}\left(1-\frac{m}{365}\right)\frac{n-1}{365}$

as only the last person in can share a birthday with those already in. This sum equals 24.62. Since the “magic number” is 23, this may sound wrong but the median attached to the above distribution is truly 23. My R code is

q=rep(1,a<-366)/365
for(n in 3:a)q[n]=q[n-1]*(1+1/(n-2)-(n-1)/365)
sum(q[-1]*(2:a))


### 2 Responses to “birthday paradox, one by one”

1. R has the neat pbirthday function, which does exactly what you think it does. Unfortunately it isn’t vectorized. You can get the same answer with
n = 1:365
sum(n * (1-sapply(n-1, pbirthday)) * (n-1)/365)

• Ah, how neat!!!

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