February 3, 2020

## A fun ice-breaker exercise

A few weeks ago my friend and colleague, Jason Forrest, asked if I would fill in at the last minute for an ailing speaker at the New York Data Visualization Society Meetup.

Seeing that the meetup had drawn a nice crowd, a few minutes before the presentation I asked Jason if I could commence with a fun ice-breaker exercise. In this exercise I assert that although it is counterintuitive, in a group of 30 or more people the chances are around 70% that two people will share the same birthday.

As I looked out at the group there were already 40 people in the room so I thought “this will be a lock!”

Please realize I’ve done this “trick” dozens of times and had even won a couple of dinner bets with as few as 30 people, so I was sure it would work out with 40 people.

Not this time.

Here’s a quick recap:

I started by asking “who was born in January?” and a few people raised their hands. I then asked them to tell me on what day in January they were born.  No match.

February?

No match.

March?

Nope.

The furthest I had ever gotten previously without a match was November, but this night we hit December and no matches.

I then started my presentation and as people entered the room – late for my presentation, how dare they! – I would look at them with hope and anticipation.  Maybe this person would restore my credibility…

It was not to be. By the end of the evening we had a total of 50 people in the room and despite a 97% probability, nobody shared a birthday.

I’m glad I didn’t make any bets!

## How to compute this

I suspect you are thinking “there are 366 possible birthdays and you have 50 people, so the chance of a match is 50 out of 366” or something like that.

Here’s how we can compute the odds.

Instead of computing the probability of two people sharing the same birthday, let’s compute the probability of NOT sharing the same birthday.

With only two people the chances of not sharing the same birthday are excellent:

365 / 366

(Yes, we’re taking February 29 into account).

What happens with three people?

(365 × 364) / (366 × 366 × 366)

(365 × 364 × 363 ) / (366^4)

When we get to 30 people, we have 365 × 364 × 363 × etc., going down to 326 in the numerator, and 366^30 in the denominator. The likelihood of it not happening reaches 32% so the likelihood of it happening is 68%.

With 24 people the chances are better than 50%, and with 50 people the chances are just under 97%!

(As for whether I should be using 365 or 366… it should probably be 365.25. I’ll leave it for others to argue.)

## Try the interactive version

Want to “see” the odds at work and get a sense of how unlikely my loss was? Click here.

Note: the workbook is downloadable and takes advantage of Tableau’s upcoming animation features. There’s also a bar chart version in the workbook.