Everyday numbers don’t scare me. The day, the date, the time are important and simply communicated. I can throw a couple of round numbers at anyone, and they should know what’s happening. Yes, convention does matter. Standards matter. I don’t know how, but I know some people struggle with the 24-hour clock notation.
When we get to small scales and tiny numbers, less familiarity means that it’s not so easy to communicate. To make those numbers meaningful media people like to use analogies. A common one is saying that a thing is: less than the width of a human hair. If you still have it, and I do, hair is an everyday item.
Let’s say a human hair is typically 100,000 nanometres wide. Sounds big in nanometres. That’s a tenth of a millimetre. Now, I can get a plastic ruler and visualise that size. My perception of scale depends on where I put the decimal point. Remember in SI Units a “nano” is 1 x 10-9[1]. Something to think about when seeing newspaper headlines about nanotechnology.
Visual depictions do help. Even if they can be slightly misleading when comparing dissimilar objects. Our planet, Earth is about 12,756 kilometres in diameter. So, for a bit of fun I could say the Earth is about 128 x 109 times wider than a hair on my head. Nice but not so useful. Tiny probability numbers like the range from 1 x 10-6 to 1 x 10-9 require some imagination.
It’s not such a big leap. Let’s say that I make mistakes. That said, I’m well trained at a specific simple task. Flicking a switch at the right time. My measured error rate is about 1 in 100. However hard I try, I make mistakes, not necessarily the same one, but with a reasonably quantifiable average frequency when nothing changes.
A well-designed machine, doing the same mechanical task, can do better than me. It’s measured error (or failure) rate is about 1 in 10,000. That might be considered good if it’s merely to switch on a toaster at precisely 6 am. It might not be so good if the result of a single mistake is instant death. In other words, I’ve become highly dependent on this mythical machine.
To do better, I could devise a means of checking the results of this machine. If I did this checking perfectly, entirely independently and without distraction, then experiencing a negative result might get up to a rate of one in a million. With this arrangement, I’m still not happy enough to place my life, or the lives of my colleagues in the hands of such a system.
Instead, I’ll construct two entirely independent well-designed machines, each doing the same simple task and each constantly checking the other one. Now, I’m cooking on gas, as the expression goes. Will this result in a negative outcome rate of around 1 in 1 x 108? One in a ten million. At least it’s an analysis worth doing. However, calculations may not give the result as one in a ten million. That result can hinge on the notion of what is entirely “independent”.
To make my general point here I have grossly oversimplified a problem. What I hope I have conveyed is that tiny probability numbers can be grasped without entertaining rocket science or nuclear physics. In the world of computational systems, we can make machines that are exceptionally good at performing consistently, persistently and error free. Not perfect. Not at all. Not prefect in so much as making life and death decisions.
[1] https://www.nano.gov/about-nanotechnology/just-how-small-is-nano
John Vincent 