Imagine a twenty-sided die (a d20 as it is known to gamers) being rolled. In the ideal the die has a 1 in 20 chance of rolling a 20 (or any number). It is natural to think of the die as being a locus of chance, a random number generator whose roll cannot be predicted. While this is an appealing view of dice, there is a question about what random chance amounts to.
One way to look at the matter is that if a d20 is rolled 20 times, then one of those rolls will be a 20. Obviously enough, this is not true. As any gamer will tell you, the number of 20s rolled while rolling 20 times varies. This can be explained by the fact that dice are imperfect and roll some numbers more than others. There are also the influences of the roller, the surface on which the die lands and so on. As such, a d20 is not a perfect random number generator. But imagine there could be a perfect d20 rolled under perfect conditions. What would occur?
One possibility is that each number would come up within the 20 rolls, albeit at random. As such, every 20 rolls would guarantee a 20 (and only one 20), thus accounting for the 1 in 20 chances of rolling a 20. This seems problematic. There is the obvious question of what would ensure that each of the twenty numbers were rolled once (and only once). Then again, that this would occur is only a little weirder than the idea of chance itself.
But a small number of random events (such as rolling a d20 only twenty times) will deviate from what probability dictates. It is also well-established that as the number of rolls increases, the closer the outcomes will match the expected results. This principle is known as the law of large numbers. As such, getting three 20s or no 20s in a series of 20 rolls would not be surprising. But as the number of rolls increases, the closer the results will be to the expected 1 in 20 outcomes for each number. So, the 1 in 20 odds of getting a 20 with a d20 does not seem to mean that 20 rolls will guarantee one and only one 20, it means that with enough rolls about 1 in 20 of all the rolls will be 20s. This does not say much about how chance works beyond noting that chance seems to play out “correctly” over large numbers.
One way to look at this is that if there were an infinite number of d20 rolls, then 5% of the infinite number of rolls would be 20s. One might wonder what 5% of infinity would be; would it not be infinite as well? Since infinity is such a mess, a more manageable approach would be to use the largest finite number (which presumably has its own problems) and note that 5% of that number of d20 rolls would be 20s.
Another approach would be that the 1 in 20 chance means that if all 1 in 20 chance events were formed into sets of 20, sets could be made from all the events that would have one occurrence each of the 1 in 20 events. Using dice as an example, if all the d20 rolls in the universe were known (perhaps by God) and collected into sets of numbers, they could be dived up into sets of twenty with each number in each set. So, while my 20 rolls would not guarantee a 20, there would be one 20 out of every 20 rolls in the universe. There is still the question of how this would work. One possibility is that random events are not random and this ensures the proper distribution of events such as dice rolls.
It could be claimed that chance is a bare fact, that a perfect d20 rolled in perfect conditions would have a 1 in 20 chance of producing a specific number. On this view, the law of large numbers might fail. If chance were real, it would not be impossible for results to be radically different than predicted. That is, there could be an infinite number of rolls of a perfect d20 with no 20 ever being rolled. One could even imagine that since a 1 can be rolled on any roll, someone could roll an infinite number of consecutive 1s. Intuitively this seems impossible. It is natural to think that in an infinity every possibility must occur (and perhaps do so perfectly in accord with the probability). But this would only be a necessity if chance worked a certain way, perhaps that for every 20 rolls in the universe there must be one of each result. Then again, infinity is a magical number, so perhaps this guarantee is part of the magic.
PPS. oh gosh, I just remembered another SUPER, SUPER weird ‘chance’ story that happened to me when I was 20 or so, more than 30 years ago. I kid you not, it seems straight out of fiction, but I already wrote a long comment and don’t want to take your time further. Eventually, in your next blog about chance, sometimes in the future! This weird stuff also happens to me on a small scale, for example I have an mp3 library with almost 10.000 different songs playing randomly, the other day I was watching an old tv series or something, and as soon as it ended and I turned on the mp3 player, wham, that tv theme starts playing. The only instance in 10.000 songs. Sure, it’s not an impossible probability, but – in conjuction – to exactly the moment when I finish watching an episode of the tv series? Just weird.
Ah ah….it reminds me of another time, I was strolling the streets without a penny in my pockets, thinking, ‘oh wow, if only I’d find some money on the way, I’d buy myself some chicken’, and wham, at that -exact- moment, I come across a £ 20 bill on the pavement. It was even raining, so it was soaked in rain, I had to go back home and use an hairdryer…
Just weird! I’ll better go….
Chance always seemed to me very hard to figure out. I had read somewhere that in tossing a coin, ‘it doesn’t matter’ how many heads or tails happened before the next toss; according to these people, a new toss kinds of ‘resets’ the whole game, but to me this seems a fallacious conclusion: surely – if one sees the whole game as ‘singles’ – , the singles probably apply, but in the context of multiple tosses, the singles seem to have a definite role as a whole: if you had to gamble, would you go ‘well, we had twenty consecutive tails, but since the previous tosses don’t matter, I place my bet on another tail’. You’d probably bet on heads, because the fact that twenty tails ‘happened’ before, do seems to have an influence on chance for this game – as a whole -. It’s almost as if we are seeing something in a narrow way vs. ‘the big picture’.
As anecdotal evidence that chance can be really weird: more than once I encountered someone I had known years before, in a city distant from where we grew up, which was a small home town in the middle of nowhere in Italy. Sure, many people leave their hometown to go live in bigger cities. But I only met again that particular person, once, for example by walking past a bus stop. If I had happened to pass by 30 seconds before or after, I had never seen them again.
Another time, I did return for two weeks to my hometown, after almost 30 years, and an uncle who had cancer for many years, died exactly within the two weeks I was there. I could not help thinking: ‘of 30 years, these two particular weeks I happen to be here, have been the most decisive for this man’. We weren’t close or anything, in fact we were little more than strangers.
And the weirdest one of all: in 2012 I damaged my knee, (exactly the time I came across your ‘Of Tendon & Trail’). I was dragging myself around, alone, in my apartment by pushing an office chair which had wheels, for months while waiting for my operation (I could not pay for it as I didn’t have a penny, so I had to wait for the NHS to sort it – I am in the UK – which fortunately did). I didn’t particularly suffer physically: fortunately there was no pain, and I could get all my groceries delivered at my address. My rent was getting paid through welfare benefits. Of course I was scared, etc, but in retrospect, it could have been worse.
Well, out of the blue, my sister sends me an email that went something like: ‘Hey. Are you allright? I ‘sensed’ something weird….’. That is all she wrote. I could not believe it, my jaw dropped, because this was completely out of character: she rarely ever wrote to me, although we always kept in touch from time to time, but really only to send birthdays greetings and the like….in short, each was too busy living their life. Of course, I made no mention to anyone about my difficult situation, something each of us actually always did, as we never wanted to trouble the other in any ways; even though ‘life’ had broken our family, we still loved each other, we just weren’t living together anymore. You know, that stuff is nothing so uncommon, although of course each of us is a unique individual with a unique story.
Upon reading, I thought to myself: ‘If I tell her what happened, she’ll live the rest of her life in damn paranoia……she’ll always be getting the feeling that maybe she’s ‘sensing’ something (whatever that means).’.
Well, I wanted to destroy the chance of THAT happening. So I wrote back: ‘Heeeeeey! How you’ve been? Hope all is ok. And, just what the heck are you talking about? I am totally fine, ha ha. Hugs.’.
I believe I am a ‘rational’ person (what a lame and common thing to say….), but this was one of the weirdest things that happened to me. To me at least, it seemed the chance of 1 in a million, so to speak.
PS. I have been curious about your penchant for these tabletop games, of which I knew nothing about. I checked out many RPG videogames through the years and invariably I find them boring. Maybe I should try these tabletop versions. Anyways, sorry for the long comment, and feel free not to publish it, to me it doesn’t matter who else reads it, just that you did so. The knee stuff and chance stuff combined, seemed to prompt my story to be told. I haven’t said this stuff to anyone else. Best wishes, Luigi.
Alright. I know there are folks who, one way or another, make their living from chance, probability, etc. Insurance companies and their actuaries do this, everyday. That form of gambling is quite legal—no one forces us to buy insurance, but, if we don’t, everyone suffers. I think this is roughly, usury, along with things like interest on money borrowed…caveat emptor, etc. Jesus, if we believe what is written, was not popular among thieves. So, OK, how exactly did we get from Jesus kicking moneychangers’ asses to other forms of usury being tacitly blessed? hmmmm…this is curious, isn’t it? Well, times change, morals morph and contingency evolves…sorta. I read today that James Earl Carter died, at age 100. I do not know that he was an honest man. Neither do I know he was not. Maybe anyone who hated him had some reason. I might think those who loved him had better ones. He never did anything to hurt me or mine, so far as I know. Chance is fickle; unpredictable—synchronistic—Jung thought so. That is metaphysics—a wild ass guess. Things happen that are inexplicable. Sure.That is the point. I was never money-changer or loanshark. Am good with that.
Infinity is distant, unimaginable. So much so that one cannot get there. There is no *there* to which one might arrive.