Birthday Triplets, Common Birthdays and Poisson Distribution

Suraj Regmi
4 min readJan 26, 2019

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In the common birthday article of Bale and Busquets, we discussed why their common birthday was a probabilistic event rather than a mere coincidence. Digging the problem further, we discuss three persons having common birthday here.

Assumptions

  • There are 365 days in a year.
  • All the days of the year are equally likely to have a birthday event.
  • The people are randomly selected so the sampling distribution is not biased.

Landscape

We discuss here the existence of at least one common birthday in at least three people. We find the probability of such common birthday existence by using naive definition and then extend the problem to Poisson distribution case. That helps us to visualize Poisson approximation and how it can be used to model real-life situations.

Naive Way

In a group of n people, there are C(n, 3) triplets. C(n, 3) is the number of combinations of 3 people among n people.

For at least one triplet to have the same birthday, let’s find the probability that no triplet has the same birthday. For that let’s take a random triplet and find the probability that it does not have the same birth date.

For finding the probability of birthday triplet not having the same birth dates, let’s subtract the probability of same births birthday triplet from 1.

So, it would be:
1- (1 / (365 * 365))

The probability of same births birthday triple becomes 1 / (365 * 365) following that, for an arbitrary person, it is probable with (1/365) * (1/365) probability that the two persons have the same birthday.

This is for a triplet not having the same birth dates. There are C(n, 3) triplets and for all of the triplets to not have the same birth dates, the probability becomes:
(1- (1 / (365 * 365))) ^ C(n, 3),
as the triplets are independent events.

So, now to have at least one same-birth triplet, the probability becomes:
1 - (1- (1 / (365 * 365))) ^ C(n, 3)

Let’s plot the graph of probability versus n and see how the probability plots with respect to n.

Graph of probability (p) versus sample cardinality (n)

For values of n less than 50, the probability is low but once the value of n crosses 50, the probability seems to increase steeply and again the rate of change of increase decreases with more increasing value of n.

At n=100, the probability becomes 0.7029. So, in a group of 100 people, it is likely that there is at least one birthday common to three people.

Poisson Way

The probability of same-birthdays triplet is 1 / (365 * 365), which is very small, and there are C(n, 3) triplets, which is a big number for the moderately high values of n (~>20). The high value of no of intervals and low value of the probability is the necessary condition for the distribution being approximately Poisson. Similarly, the events are weakly dependent so that it is reasonable to model the scenario with Poisson distribution.

Now, the parameter lambda is expected value of the distribution which can be calculated as:
no of triplets * probability of same-birthdays triplet
= C(n, 3) / (365 * 365)

To calculate the probability that there are no such events, let’s substitute the value k=0 in the Poisson distribution.

P(X = k) = e^(-lambda) * (lambda^k) / (k!)
P(X = 0) = e^(-lambda) * (lambda⁰) / (0!)
P(X = 0) = e^(-lambda)

For at least one such event to happen, let’s subtract P(X=0) from 1.

So, the probability becomes:
P(X ≥ 1)=1- e^(-lambda)

Substituting the value of lambda,
P(X ≥ 1)=1- e^( - C(n, 3) / (365 * 365))

Put n=100 and we would have, P(X ≥ 1)=0.7029.

Both the ways, the probability is the same for n=100 i.e 0.7029.

Now, let’s compare two probabilities and see how our Poisson approximation compares for smaller values of n (<15).

Comparison of actual probability and Poisson approximated probability for n < 15

The approximation is awesome and it is correct to at least 6 decimal places for small values of n.

Of all the discrete distributions in Probability and Statistics, Poisson distribution is arguebly most important distribution. It has its applications varying from telecommunication to astronomy, biology to radioactivity, etc.

Something special to explore for all the distribution and pattern lovers out there!

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Suraj Regmi
Suraj Regmi

Written by Suraj Regmi

Data Scientist at Blue Cross and Blue Shield, MS CS from UAH — the views and the content here represent my own and not of my employers.

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