Keno, lotteries and very large denominators
Posted 1 April 2007 at 17:41
Some games are built on numbers so large that intuition simply stops working. A draw that selects twenty numbers from eighty, or six from forty-nine, produces a count of possible outcomes that no one has any experience of. The mind treats a denominator of fourteen million the same way it treats a denominator of fourteen thousand, which is to say as a large number, and that flattening is the entire difficulty.
The count itself is easy to write down. Choosing six items from forty-nine without regard to order gives forty-nine times forty-eight times forty-seven times forty-six times forty-five times forty-four, divided by the six ways of ordering six items, which is seven hundred and twenty. The result is 13,983,816. The arithmetic is a few minutes of work, and the number is exact, which is more than can be said for almost any other quantity in this subject.
Feeling that number is harder than computing it. One way is to convert it to a length. If each possible selection were written on a card one millimetre thick, the stack would stand about fourteen kilometres high. Another way is to convert it to a duration. If one selection were checked every second without pause, the full set would take a little over five months.
Keno stretches the same arithmetic further. Selecting twenty numbers from eighty produces a count with nineteen digits, and the interesting quantities are not the count itself but the smaller counts inside it: how many of those selections contain a given five of the player's numbers, or a given eight. Those are products of two binomial counts divided by the whole, and they fall away steeply as the match required rises.
The practical lesson is small but firm. When a game's structure produces a denominator of that size, no amount of watching, timing or selecting shifts it. The denominator is a property of the draw, not of the person reading it, and the only honest way to hold it in mind is to write it out in full rather than to summarise it as a large number.
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