Enterprise has a crew of 1000 souls. 1000 / 365, means that on average 2 - 3 people share a birthday.

Plugging this into a birthday calculator, there’s a very high chance at least 7 people share a birthday on the Enterprise. Not only that, but there should only be 24 days in the year when Ten Forward isn’t occupied with a birthday party.

So given that birthday collisions are plentiful, and that birthdays are constantly happening, and that Ten Forward is the only bar in the whole damn ship… my question is: Why does Riker get to celebrate his alone with full fanfare?

Edit: Birthday Math you asked for it

  • The probability that anyone has a birthday on any given day is 1/365 and can be modeled with a Binomial distribution (Yes/No).
    • Y ~ Binomial(n=1000, p=1/365)
  • When the number of samples is large (ship of 1000 people) and the probability is small, you can model repeated Binomial distributions using a Poisson distribution (independently occurring random events at regular intervals kinda) with a mean of 1000/365.
    • Y ~ Poisson(μ = 2.74)
  • The probability mass function (area under the curve) gives you the probability of a value against that distribution, and for Poisson is PMF(Poisson) = e-μ*(μk / k!)
    • P(Y = K) = PMF(μ, k) (for K exact events, e.g. 10 birthdays)
  • So to calculate the probability of 10 birthdays in a year, you do: 365 * e-2.74*(2.7410 / 10!) = 0.155 days (i.e. zero)

The following python snippet gives you these outcomes:

Code 
import math
def poisson_pmf(mean, k):
     return(math.exp(-mean)*mean**k/math.factorial(k))

def num_days_with_birthdays(num_people, num_birthdays): 
    return(round(365*poisson_pmf(num_people/365, num_birthdays)))

souls = 1000
print(*(f"{n:2d} days with {x:2d} birthdays"
        for x in range(50)
        if (n := num_days_with_birthdays(souls, x))),
      sep="\n")

## For a ship of 1000 people in a 365 day/year cycle
24 days with  0 birthdays
65 days with  1 birthdays
88 days with  2 birthdays
81 days with  3 birthdays
55 days with  4 birthdays
30 days with  5 birthdays
14 days with  6 birthdays
 5 days with  7 birthdays
 2 days with  8 birthdays
 1 days with  9 birthdays

Edit2: In retrospect, Riker could have been one of those 65 people
Edit3: But even then! You don’t necessarily do your birthday on the exact day, but wait until the Friday or Saturday night to celebrate it, which would definitely have collisions. So Riker’s either super special, or an inconsiderate prick. There, solved it