# The almost surely probability concept

The almost surely probability concept

In: Mathematics

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Let’s say that I pick any positive integer, at random. What is the probability that the number I pick is 42?

Well let’s start with an easier question: are the odds that I pick 42 more or less than 1/10? Surely it is less than 1/10, because I had more than 10 numbers to choose from. If I did this over and over again, there’s no reason why I would pick 42 any more often than the other numbers, so I wouldn’t pick it more than 1/10 of the time. This should be pretty easy to convince yourself of.

What about 1/100? Again, it should be pretty clear that the chance that I’ll pick 42 is less than 1/100.

What about 1/1,000,000? Again, there are way more than a million numbers to pick from, so the odds that I pick 42 are smaller than one in a million.

We can keep going, as far as you want. You name any big number N, and the odds that I picked 42 are smaller than one in N.

*There is no positive probability that is smaller than the probability that I picked 42.*

I could make the same argument for any number – not just 42. Whatever number you name, the chance that I will pick it is so close to zero that it’s tempting to just call it zero.

But I am going to pick some number, so it also doesn’t make sense to assign a probability of zero to each number. We can use “almost surely” to describe situations like this, where the probability of an event is smaller than any positive number, but the sum of the probabilities of all the events is one.

There’s a practical problem with the game I described above, which is just meant to give you something to chew on:

What’s the chance that the number I pick is less than a million? It turns out that by a similar argument, I will almost surely not pick such a number. The same is true for the chance that I’ll pick a number smaller than a billion.

Even scarier, the chance that I’ll pick a number with *fewer than a trillion digits* is basically zero.

Even though there are an unimaginable number of integers with fewer than a trillion digits, there are so many more integers with more than a trillion digits that I am pretty much guaranteed to pick one of the bigger ones.

So the game is a bit silly, because just writing down my number would *almost surely* take longer than any amount of time you give me to do it, even if that amount of time is billions and billions of years. Yikes!

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.