1

From this post, the birthday problem involving more than 2 people can be approximated using a Poisson random variable. But I am wondering whether a binomial random variable can be used here. I imagine that for each person, there is some chance that they have a particular birthday and some chance to not have a particular birthday. Plus, when there are more people in the room (which makes the probability of having distinct birthdays very low), a binomial random variable will behave like a Poisson random variable.

As an example in a room of 23 people, the Poisson approximation is

$$1-\exp\left(-\frac{{23 \choose 2}}{365}\right)=0.5$$

If binomial random variable is preferred,

$$1-\left({{23 \choose 2} \choose 0} \left( \frac{1}{365} \right)^0 \left( \frac{364}{365} \right)^{{23 \choose 2}}\right)=0.50048$$

It looks close enough. My professor said that I can't use the binomial distribution to solve the birthday problem because events are not independent. Does what I found contradicts what he said?

Jimmy Yang
  • 294
  • 1
    The events are not independent, but they are "weakly" dependent, which is why the binomial can suffice and your answer is close to the actual answer. Your professor's advice is right when there are many people and weak dependence is not true, but in your case, depending on the accuracy you want (you seem to be satisfied) the binomial can work. – Sarvesh Ravichandran Iyer Jan 24 '23 at 07:52
  • Not sure what the professor actually means : Perhaps that the birthdays are not independent or not equally distributed in reality ? But in such exercises this is always assumed. The poisson approximation for a binomially distributed variable works well , if $n$ is large and $p$ small. – Peter Jan 24 '23 at 09:15
  • So which one is more preferable for approximation? Poisson or binomial? – Jimmy Yang Jan 24 '23 at 09:30
  • 1
    With an approximation, the aim is to have something reasonably easy to calculate which is reasonably accurate. In your example something like $1-\left(1-\frac1{365}\right)^{253} \approx 1-\exp\left(-\frac{253}{365}\right)$ is well known, though I would have thought the latter might be easier. Both the Poisson and the binomial approximations make assumptions which are not correct but where the impact is small with $23$ people (the relative error would be large with $366$ people since both would give you a small positive number instead of $0$) – Henry Jan 31 '23 at 14:43
  • The binomial approximation is actually slightly closer to the true figure than the Poisson approximation is. In a sense the Poisson approximation might be seen as an approximation of the binomial approximation (it allows you to have an unlimited number of pairs of people sharing a birthday when you have a finite number of such pairs) – Henry Jan 31 '23 at 15:01
  • @Peter - the birthdays may be independent, but the pairs of people are not quite: if A and B share a birthday, and B and C share a birthday, then you know A and C share a birthday; similarly if A and B do not share a birthday and if B and C do not share a birthday, then the probability that A and C do not share a birthday is $\frac{363}{364}$ which is slightly less than $\frac{364}{365}$ – Henry Feb 25 '23 at 23:58

0 Answers0