Coincidences are just waiting to happen. For all those times you've witnessed uncanny events, or had cause to utter 'small world', mathematics will explain them before you can say the words 'conspiracy theory'.
So how likely are you to share your birthday with another bloke in the pub ?
To take a famous example, consider a class of 23 students. It is more likely than not that somewhere in that class, two students will share a birthday. In a class when this is noticed, most will cry out their amazement, but the mathematicians will nod sagely to each other when they calculate that there was a 50.7% chance of this happening. (If the year group is of the right age such that some may be born on 29th February of a leap year, then we need 24 students to tip it over the balance, but the principle is the same.)
The explanation for this is that there is a large number of pairs of students upon whom the coincidence may fall. For 23 students, there are 231 ways (22×21÷2) of picking two students from the class, which means 231 potential coincidences that may occur. We expect pairs to have the same birthday once in every 365 times, so with 231 pairs it is unsurprising when one of them strikes lucky. This reasoning applies to larger situations as well. For instance, whenever a disaster occurs, we hear stories of people claiming to have had premonitions of it in their sleep the night before. To this, mathematicians say, 'Why not?' With so many major world events occurring, and so many people sleeping and dreaming, there is an enormous number of opportunities for people to happen to dream about a stockmarket crash, say, on a night before one occurs. For one person and one event the odds are long, but with nations of dreamers you can safely bet that ten or twenty of them will happen to have an apparently prescient dream on the eve of the next calamity.
We can see this type of phenomenon in everyday life whenever we watch a lottery draw. The chances of any one person winning the jackpot is minimal, and yet most weeks someone does win. The large number of players compensates for the unlikelihood of any one given player winning.
Gambler's Fallacy I was once asked by someone whether a lottery winner had 'used up their luck', her argument being that to win the lottery again next week would be such a cosmically unlikely event that their chances cannot be as high the other players, who have never yet won. In fact this is untrue; they are neither 'more lucky' or 'less lucky' as a result of their win. Once they clinch their first win, they are already half way to winning twice, and they have just the same (un)likelihood as any other individual of taking the winnings the following week. The odds that someone, somewhere, will win twice in a row are just the same as the odds of you personally winning once.
So rare events do necessarily occur. As we delve further into mathematics, though, we find even stranger paradoxes. Imagine a 'lottery' (or at least a mathematical idealisation of one) that chooses at random any point on the surface of the planet. Since there are infinitely many such points, the odds of it choosing a given point are precisely zero. And yet, one point will be chosen. This is an example of an 'almost impossible' event occurring. As a final paradox, the word 'almost', used here with apparent brazen vagueness, has a very precise meaning within mathematics, in a field called 'Measure Theory' that underpins most of probability.
Bear in mind, though, that not even mathematics can help you if you forget to buy your lottery ticket !