“Unless you expect the unexpected, you will not find it, for it is hidden and thickly tangled.”

Heraklitus, fragment #18 about 500 BC

The mathematics of serendipity

Serendipity brandmarks the frequently surprising discovery of a solution for a problem which has not been requested for, yet. This effect may be explained by the so-called “Birthday Problem”. It deals about the strangly high probability that two persons of a group share the same birthday. For mostly we tend to confound it with the probability that someone shares his birthday with someone particular – that is mostly me. And we neglect the disproportionately increasing possibilities of combinations for a probability that “any” coincidence of birthdays occurs. In a similar manner serendipity can be explained that one finds much more easily a solution for “anything” than a solution for “something” in particular.

Thus, the propability that someone shares my day of birth – neglecting leap years – is 1/365 or about 0.27%. This probability doubles to “almost” 0.54%, if I compare my birthday with two others instead of just one. It is just “almost”, because there is a propabability that the second partner has by chance the same birthday like the first one. Therefore, the probability for coincidence does not increase linearly by the number of participants, but even for any “arbitrary” number of participants a risk remains that the desired combination has not appeared, yet.

In contrast, if I want to figure out, whether “anyone” of the three of us shares a mutual birthday, then the comparison of me and two others yields already three instead of two options yielding a probability of accordingly “almost” 0,81%. For there is not just a possible coincidence with me, but also a coincidence of both others without me. Hence, these possibilities increase strongly. As there is a meager possibility of 5,9% that someone of 22 participants shares my birthday, the possibility depasses already 50% that among us 23 participants there is “any” shared birthday. And with 365 participants I am even sure that there has to be a given coincidence, although a possibility of about 37% remains that no one shares a birthday with me.

Similarly, serendipity may be regareded as the event that a occasional experience turns out to be “somehow inventive”. And consistingly there is some evidence that more than 90% of inventive ideas occur outside of a professional quest, be it working place or time. Behind that a “serendipitous” effect can be assumed providing less than 10% of success for the purposeful search of  a (specific) invention at labor time, whereas any arbitrary, general attentiveness yields more than 90% of success.