The almost surely probability concept
In: Mathematics
Think about throwing an infinitely-pointy dart at a square poster that you can measure infinitely well, to an infinite precision and accuracy. This poster is half red and half blue, cut across in a diagonal line. You throw the dart perfectly at random, and you always hit one spot on the poster: the red side 50% of the time, and the blue side 50% of the time.
What are the odds of getting your dart to land exactly on the diagonal?
Well, it’s zero. Not just super, super unlikely, like randomly shuffling two decks of cards of having them come out identically, but actually *zero* chance. The closer you zoomed in on your dartboard, the more you’d be able to see that the dart is on one side or the other, no matter how slightly. There is no point where the dart can hit the exact diagonal, because the exact diagonal has no area. It *exists* — the blue half is the blue half, and the red half is the red half, and because there’s no gradient between the two we can say it’s a perfectly sharp transition between the two — but because it has no area.
The problem is, a dart landing on one of those points is in theory no less likely than it landing on any other specific point on the board, when looked at at the same scale. Mathematicians get around this by saying ‘almost surely’: there is a probability of 1 that the dart will land somewhere other than the border, but the border is still — technically — a valid place for the dart to land.
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