Is it really a coincidence? ……….Innumeracy series (3)

photo taken at Bhigwan Lake, India by Prateek Kumar Rohatgi

“The paradoxical conclusion is that it would be very unlikely for unlikely events not to occur” — John Allen Paulos

This week, a very dear friend of mine was struck by a mighty coincidence. Last Tuesday happened to be the birthday of four different people from his social/family circles. With 365 days in a year this did seem like one heck of a coincidence. But before I get into what happened after he shared this observation, here is a question for you:

How many people do you need to assemble in a room such that there is a 50% chance that at least two of them share a birthday — just the day and the month, not the year. If dusting those “grey brain cells” that stored your knowledge on “probability” is proving difficult at this particular time, put out a random guess.

Well, the answer is just 23 people. If you have not figured it out yet, a quick google search can give you the steps to arrive at this number. This problem is known as Birthday paradox, by the way.

Coming back to my friend’s observation of rare coincidence, I had an unfair advantage of having dealt with the birthday paradox problem in the past. I sat down to work out calculation for four people sharing the same birthday. It took me about 30 mins to realize that mathematics involved here is very complex (unlike the case for two people sharing birthday) and is beyond my strictly amateur competence.

A little second order googling revealed the following information:

“ you will need…………

23 people to have at least a 50% chance of at least one set of two people with the same birthday,

88 people to have at least a 50% chance of at least one set of three people with the same birthday,

187 people to have at least a 50% chance of at least one set of four people with the same birthday,

313 people to have at least a 50% chance of at least one set of five people with the same birthday”

187 is not such a large number in terms of social and family connections for most of us. As a matter of fact this is well in the range of Dunbar number. (Dunbar’s number is a suggested cognitive limit to the number of people with whom one can maintain stable social relationships — generally (erroneously) considered to be 150 but proved to be a range from 2 to500. — from Wikipedia)

So as it turns out, what seemed like a big coincidence is after all not such a big one.

  • Coin flips: I tried an experiment with coin flips (simulated on computer to avoid biases from my flipping technique). Five sets each comprising ten coin flips. The ratios of heads to tails for these sets came out to: 0.43, 0.67, 1.5, 1.5, 2.33. Whereas if I pooled these observations cumulatively, the ratios were: 0.43, 0.54, 0.76, 0.90, 1.08. For smaller sets of observations, the ratio was hardly ever close to 1 whereas as the number of coin-flips increased, ratio of heads to tails started to approach 1 (expectedly). Importantly, even a fair coin does not throw up fair results except for larger sets of flips.
  • Astrology: most pop astrology works only because astrologers predict a large enough number of non-specific possibilities for the future such that mostly one or more of these predictions are quite likely to turn out true for someone or the other.
  • Stock market predictions (I perhaps could have clubbed this with astrology): Keep predicting a crash all the time and it will turn out true once in a while and when it turns out true (and assuming you made sufficient public display of it), you may catapult into a stock market seer status overnight. A few who get the predictions right more often than wrong but even they do not predict “when” — they only predict “what” and patiently wait for the “what” to materialize — many times for years on end.
  • Personal connections: Once on Facebook, I was surprised to see my wife’s cousin to be friends with an ex-colleague of mine — it felt like a coincidence but it did take much to realize that such connections (the friend-of-a-friend coincidences) are not uncommon. And of course there is the concept of “six degrees of separation”.

So why do so easily yield to coincidences: one reason definitely is that probability is a beguilingly non-intuitive subject. Another perhaps is an innate need in most of us to discover magic. It is somehow important for us humans to see bigger purpose or magic in things we do or involved in or come across.

It can be very valuable to separate luck from expertise. For example when on a search to engage an investment manager, it is important to assess contribution of luck (coincidence) and of expertise on the past performance of the manager. Separating ability from coincidence can prove to be a significant factor in long term success of any endeavour.

Ironically, among the most coincidental facts is one that we are alive at this very moment. It takes just one thing among an infinite possibilities to cause us to cease living but it takes a whole lot of things to come together to be alive. However it is death that we take ominously notice of but not the miracle of being alive.

Interesting items I came across:

  • The book “Innumeracy: Mathematical Illiteracy and Its Consequences” by John Allen Paulos: To the extent that I read it, it is an excellent read — perhaps the most popular book on innumeracy
  • The pigeonhole principle (also called Dirichlet’s Box Principle): If there are more pigeons than pigeonholes then when pigeons fly into the holes, at least one of the holes will end up having more than one pigeon. Before you say ‘duh!’, this principle has good application in probability — lot of material on the web — couple of immediate references:



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