Although we know that the probability of an unspecified “unexpected” event is rather high, given how many possible events can occur, it is still interesting to note them. There is probably an evolutionary reason why probability is so unintuitive to us. Perhaps being curious about coincidences had substantial survival value in the past.

Here are two that have occurred to me recently.

1) Just over a week ago I wandered into the Mechanics Institute Chess Club (apparently the oldest in the US) in downtown San Francisco.There were just a few players sitting around casually on a Sunday afternoon, with no events scheduled. Then in walked someone I haven’t had any contact with for 25 years, whom I knew from playing against in the schools tournaments in Christchurch and at the Canterbury Chess Club, who later became NZ champion. Now that is a coincidence – he works for Google in Sydney and was in SF for a conference. Neither of us has played competitively for many times longer than our competitive chess career lasted. Playing a few games with the newfangled clock (3 minutes per player, plus 2 second increment per move) was a fun way to spend an hour.

2) This is close enough to true, and doesn’t change the essential mathematics. I know someone who has been married twice, each time to someone with the same birthday. How likely is that? This is different from the famous “birthday problem”. Suppose that everything is uniformly randomly chosen, and I have $latex k$ acquaintances who are in a position to marry and whom I know well enough that I would hear about such a coincidence. Let $latex n$ be the number of days in a year. The probability that one of these acquaintances fails to have such a coincidence is $latex 1-1/n$, so the probability that some succeeds is $latex 1 – (1-1/n)^k$. Let $latex c$ be a number between $latex 0$ and $latex 1$ that represents our threshold for incredulity – if an event has probability less than $latex c$, I will be surprised to see it, and otherwise not. Thus we should be surprised if $latex (1-1/n)^k > 1-c$. Reasonable values for $latex n,k,c$ are $latex 365, 100, 0.01$, but since $latex (364/365)^{100} = 0.76$, we should not be surprised. Analytically, the inequality $latex (1-1/n)^k > 1 – c$ can be approximately solved via making the approximation $latex (1-1/n)^k = exp(k log(1-1/n) approx exp(-k/n) approx 1 – k/n$. Thus I expect to need $latex k < cn$ in order to be surprised: the number of acquaintances should scale linearly with $latex n$, which clearly doesn’t have to measure days. So if I only care about which month the person is born in, I will never be surprised, but if I care about the hour and have fewer than about 80 acquaintances, very likely I will be surprised. The original case of a day shows that if I have more than 4 acquaintances, I should not be surprised.

Suppose everyone in the world is my acquaintance. How precise could we be about the birthday without being surprised? Now $latex k$ is of the order of 5 billion, so with the same value of $latex c$, $latex n$ should be about 500 billion. That means we can slice up a year into milliseconds and such a coincidence event would not be at all strange.