trI.m in the midst of writing the chapter on ‘the best laid plans of mice and men’. It deals with how the unexpected steals up the the tragic protagonist. Uncertainty, risk, unexpectation (is that a word?–now it is!), and things like that are on my mind. One way of imagining risk would be to graph outcomes onto a bell curve. The fat tails on the extreme left and right sides of the curve could represent unexpected disaster or a happy windfall. Another way of imagining risk would be look at dice or card games.
We.re surrounded by so much probability theory and statistics today that it.s hard to imagine a world without such things. But the science of probability or a theory or permutations and combinations didn.t actually exist before the likes of Cardano and Tartaglia started systematically going through how many outcomes were possible when rolling one die, two die, and so on. That was as recent as the Italian Renaissance in the sixteenth century. Before then, how the dice turned out was all due to Lady Luck, otherwise known as Fortune. If you could go back in time with today.s probability theory and play the ancients, you.d be able to clean house. The odds on a lot of the ancient games rewarded higher outcome scenarios more than lower outcome scenarios. Cicero and Aristotle both thought about ‘likelihood’ and all they could come up with was that it would be hard to roll more than one or two ‘Venus throws’ (the highest throw with knuckle bones) in succession. It didn.t occur to them that such things could be quantified. They were, however, express scepticism that the ‘Venus throw’ would be due to the action of the goddess. But they were not able to offer a better explanation.
Surprising. The ancients gave us geometry, the Hippocratic Oath, democracy, philosophy, ethics, and so many other things but they just could.t get probability. Some say it.s because the dice they used were inconsistent (being polished animal bones). Others say the idea of the hand of god in random events was too powerful for the mind to overcome: the whole industry of divination was based on finding meaning in random events that, well, were not really random but god trying to tell us something. There are those who think they just didn.t have the mathematical capacity with their cumbersome roman numerals. Or they just didn.t like ‘experimenting’ (ie rolling hundreds of dice and recording the results).
That could all be true. But even today, it.s hard to figure out how the theory of combinations and permutations fit together. Last night, I was over at TW.s. As he took out some playing cards, he said, ‘Did you know the chances are that a deck of cards has never been shuffled with the cards in the order the are in now?’. I said, ‘Really?’. He replied, ‘There.s almost an infinite number of combinations so that you.d never in an eternity shuffle the cards into the same configuration’. TW.s into science so I knew he was right. But I was curious. How many combinations were possible?
We couldn.t figure out all the combinations of the 52 card deck. But we could try figuring out the combinations of one, two, three cards and so on. And from there generate a rule to see what the combinations would be for a full deck. With one card there.s one combination. With two cards there.s two. With three cards, we couldn.t do this in our head anymore. So we laid out the cards. Six combinations are possible with three cards. Now with four cards, it gets tricky. Not only did we need the cards in front of us, we had to start writing down the combinations since it was easy to miss one or count one twice. The combinations get bigger very quickly is what we noticed. I was thinking the pattern would be 1 card 1 combo, 2 cards 2 combos, 3 cards 6 combos, and maybe 4 cards would be 16 combos. Wrong. 4 cards is 24 combos. We speculated on the pattern. Maybe you multiply by a number 3×2=6, 4×6=24. But what sort of rule would determine the multiplier? The clear thinking beer we were imbibing was also helping our efforts! So we decided to work out the combinations for five cards to see if more data would lead to an insight (Bacon.s method of induction). But with five cards there were so many combinations… Too much work, we went back to drinking beer and watching a TV show on science instead. But this goes to show, it.s still difficult today to figure out probabilities. TW.s a project manager so he.s good at numbers. Years ago (certainly not today!) I got up to second year calculus.
So I cheated. The next day I googled it. Google is also something that Cicero and Aristotle didn.t have! The combinations are a function of factorials. So four factorial or 4! will give you the combination of four cards. Four factorial would be the equivalent of 4x3x2x1 or 24. Five factorial or 5! or 5x4x3x2x1 or 160 is the number of combinations with five cards. I would hate to even try imagining how big the number 52! generates. It would likely break a gear in the brain. So perhaps this is one of the reasons cards are fascinating: the unexpected is always possible because of the immense number of outcomes that are possible. Or–lurking underneath all the ‘common’ poker hands (full house, pair, two pair, etc.,) there.s always the chance of the dead man.s hand!