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Entries in probability (2)


The Birthday Problem amongst Friends

I continue to be fascinated by the Spooky Math of Coincidence that we have discussed in a previous blog post.

Within this particular branch of probabilities, we have the classic Birthday Problem: What is the % likelihood that two or more people share a birthday in a group of 30 people?

Well, I decided to see how things work on a large sample of people and dates - that being my own Facebook account. It occurred to me as I was writing “Happy Birthday” on people’s walls that sometimes there are two or three people with birthdays on the same day. So I decided to plough through the list and see what I could find out.

Out of 544 friends on my Facebook account, 422 have birthdays recorded (78% of all friends).

  • On 113 / 365 days (31%) no one in the sample had a birthday.
  • On 131 / 365 days (36%) just a single person had a birthday.
  • On 79 / 365 days (22%) two people shared a birthday.
  • On 36 / 365 days (10%) three people shared a birthday – that being three “pairs”, i.e. each of these people having a match with each of the other two.
  • On 5 / 365 days (1.4%) four people shared a birthday – that being six “pairs”.
  • On 1 / 365 days (0.3%) five people shared a birthday – that being 12 “pairs”.


Pie chart of birthday distribution 

So on 33.7% of the days, at least two people shared a birthday.

It seems that there are some days that are more "popular" for birthdays than others. This real-life phenomenon is not taken into account in the classic problem which assumes that all days have an equal probability.

Unfortunately my math is not quite good enough to do the fun stats, like taking different groups of 30 people amongst my friends and looking at their matches. I am looking at using freelancers on various web sites to help me out. I’m also posting my spreadsheet here so if someone fancies having a go at the data set, they are welcome!

In the meantime, I will make sure I remember to say happy birthday to someone in my group at least once every third day.


The Spooky Maths of Coincidence

If there are thirty people in a room, what is the percentage likelihood that two or more people share a birthday?

Less than 5%? ... 10-25? ... 25-50%? ... over 50%?

If you are like half of the population (well, at least like the 130+ executives and investment professionals who gave this answer in our LinkedIn poll as shown below), you would vote for less than 10%. That is what our instinct tells us. If my birthday is the third of March (which it is), that's 1/365 odds, surely...

Orcasci Birthday Problem poll results - by answer

Orcasci Birthday Problem poll results - by job function

And yet, this ego-centric view of the world is actually completely and utterly wrong - by an order of magnitude. The correct answer is …71%! And you only need 23 people in a room to get a 50:50 chance of a match.

The Birthday Problem Logic

When we intuit our answer, we start from our own ego-centric view and extrapolate. This, though, is the wrong mathematical approach for solving the problem. Instead of it being a "small math" problem, it is a "large math" problem. We need to work out for each member of the group the probability of a match with the remaining members. Instead of dividing the odds, we need to add them up (for more info, check the Wikipedia entry on the Birthday Problem). And counter-intuitively we find that the spooky maths of coincidence takes hold at scale.

I have been intrigued for a long time by the weird effects of numbers at scale, particularly on the human level when lots of people gather together in some way. My work on Collective Intelligence and Idea Management at Imaginatik was based on the practical application of this insight. And at Orcasci we are taking it to the next level in its application in the Science of Spread™.

One of the techniques we have used to scale our activities, starting with personal, one-on-one connections, exploits the strange nature of coincidence. An example would be going to a conference and telling twenty strangers that you have a plan to "change the regime of North Korea for better".

What are the odds that any one of the twenty people would be able to help (as opposed to laughing or switching topics)? The odds of a match are astronomically low. The same is true if you ask a hundred people - low, low, low. You can boost it dramatically in certain ways, though, such as by going to a conference on North Korea. To some, that sounds like cheating the system. Well, so what? If your end goal is making a connection, then you change your tactics to "fish where the fish are".

The Coincidence Problem of 10 Conversation Domains

Let’s look at another example of coincidence. Imagine that each of us has a set of, say, ten "conversation domains". They might be e.g. where you were born, your job, your education, where you live, where you are about to travel to or have just come from, and what hobbies you have.

Within each domain, there will be hundreds and thousands of options. For example, there are over 200 countries. And there are many, many professions.

An individual will therefore have a "life set" of options, clustered into these ten domains. That could be several thousand total options, out of a total world set of millions.

The Coincidence Problem can now be stated in two versions:

  • The first version is: What are the odds that two people, selecting ten options from their life set each, find a match?
  • And the second version: In a group of, say, five people, each person talking to each of the other four, what are the odds that any pair match takes place?

It intuitively seems as though the odds of the first version being true should be low. I'm struggling with the actual probability maths on both of the problems, mainly because I am weak on some aspects of Quantitative Methods I seem to have skipped at university. (If you personally can help - or know someone who is strong in probability and would be willing to help ;) - can you please email me, Mark Turrell?)

It is not hard to recall your own experiences of going to a party or networking event and sharing ten such items. Admittedly you might not be listening to the other person, and some people have a slightly annoying habit of only sharing their own list, adding a bunch more, and then not listening to the other person. Still, in an evening you quite often come home amazed at the coincidences that emerge.

One important point is that many options and domains are connected. For example, a country might be your country of birth, where you studied, where you have just visited, where your partner is from. That makes "country" or locations in general a pretty easy option to key off. In this concept, one option opens up a new set of related options, all of which increase the odds of finding a match.

Some options are clustered together. In this, if you do one thing, you probably do several other related things. Again, this boosts the odds of a coincidence match.

A high proportion of people are psychologically programmed to like making connections, and most people go out of their way to find associations. This encourages people to find a match, even if the link between options might seem slight.

One can imagine developing this theory further to consider strong and weak matches. A strong match would be a 100% connection between what is expressed by the parties as their option, such as "Did you do your MBA at XYZ school?" - "Yes! That's amazing!" A weak match is less content-specific, such as "I also did post-graduate studies". It is easier to find a match with weakly defined options, and deeply spooky on the occasions when you meet someone with a strong match.

Sales people and consultants often get training on how to prompt this matching effect. As people prefer doing business with people they can associate with, these connections become very important for potential business, and the faster and easier one can make connections the better.

In my personal experience (and I get out and talk to people a lot) the matching rate is about 35%, that being finding a match in 1 in 3 interactions. I am sure that my stats are skewed, because I like coming up with near-random-seeming options that provoke, startle and help me stand out from the crowd (it's a handy tactic). More often people stick to the basics of communication, covering classic topics of job, birth place, holidays, etc. On balance, without more research, I'd be comfortable thinking that the odds of any two people finding a match are between 15-25%.

In terms of the second version of the Coincidence Problem, which looks at a group of five people, this involves taking the probability of the first version, and doing the "birthday problem maths". The odds of any pair match being created are very high, so much so that a group of people cross-sharing will find a large number of matches.

Where this then becomes very interesting is trying to force very weird connections being made. What are the odds that the person you are talking to has produced a movie, has dissected a brain or knows Chinese?

What you learn for your next networking event, and it is a sure-fire tactic, is that you need to share ten things with lots and lots of people. And encourage others in a group to start sharing.

The Orchid Model: Helpers

A final thought - and there is a lot more detail we could get into, but for the sake of brevity we'll chop it off here. Some people are naturally inclined to be connectors. In my work at Imaginatik on the importance of personality styles in innovation, the model I developed, called the Orchid Model, recognized that around a quarter of the population are "Helpers", social connectors.

To boost the odds of having connections made, connectors are fabulous as they open up their network of remembered contacts and possible connections for you, without requiring the effort of having to meet the non-connected folk.