The Birthday Probability Paradox, Subsea Cable Outages, and Russian Sabotage

In a class of 23 students the probability of at least two having the same birthday is 51%. In a class of 30 students the likelihood rises to 70%. I was introduced to this counterintuitive result in my first graduate school probability course. There were no more than 15 students in the class. Yet  I and another classmate is that course shared a common birthday of March 5, 1962. I will give you the logic behind this result at the end of the post. 

This probability calculation is a clear warning human intuition is not likely to be a good guide to whether an event is natural or man-made or a conspiracy. Most people will assume that the chance of a two or more students with the same birthday is 10% to 15%. In case of telecommunications we have at least 450 subsea fibre optic cables in service today. This means that clusters of cable outages like we see now in the Baltic Sea and earlier this year in the Red Sea and off Abidjan are virtually guaranteed to happen due to pure chance alone. The likelihood that outages occur in a steady flow over time converges quickly to zero. This is convergence in the mathematical sense. Random chance does not mean as most people assume that events happen in a smooth fashion. Instead, they will often be lumpy or cluster together. 

I don't know whether the recent spate of Baltic Sea outages is sabotage or not. But probability emphatically tells me that chance alone guarantees multiple outages right on top of each other. None of this should be surprising to you. Everyone of us has experienced unusual coincidences in their lives. For example, I was walking down 73rd street in 2001 on the Upper West Side in Manhattan one day and ran into a girl I had known ten years earlier in Iowa City, Iowa. I also encountered on the street a group of 4 people from Iowa City whom I had known extremely well when I lived there when I was down in lower Manhattan one day. Is God telling me I should return to Iowa City? I hope not. The point is that given enough time unusual events are guaranteed to happen. In fact, it would be bizarre given the long length of a human life if such events did not befall us. 

Right now we have a bunch of armchair military analysts and high ranking European government officials panicking over the Baltic outages who might want to open their university probability book and brush up. Particularly in light of their rush to judgment. In fact, given the huge increase in Russian shipping through the Baltic Sea since the war started the likelihood of suspicious but entirely random outages converges to one. Cognitive bias, like death, spares no one regardless of degree or title. 

The birthday result b the way can be proven as follows. It is an exercise in combinations. Let's calculate the probability of 15 unique birthdays. There are 365 possibilities for the first student. So he or she is guaranteed a slot. So we can assign a 365/365 probability to him or her. The second student has 364 available dates. So his or her probability is 364/365. And so on. To obtain the probability of 23 people having unique birthdays, you multiply together these probabilities. The answer is 49%. So the likelihood of at least two people sharing a birthday: 1 - Probability of None Sharing a Birthday = 1 - 49% = 51%. QED. 

As physicists have pointed out, if the universe is really big, lots of strange events that you and I regard as unlikely are almost certain to happen. In fact, in an infinite universe, everything does happen including an infinite number of Roderick Becks (as scary as that prospect may seem to some of you), pink zebras dancing down main street, and convicted felons become President of the United States. If it has not happened yet, just wait a little longer.😃