Sunday, January 24, 2016

Big numbers, big confusion; small numbers, bigger confusion.

I can make something almost impossible happen. And so can you.

Let's start by defining what "almost impossible" means. Less than one-in-a-trillion chance? How about less than one in a trillion-trillion chance? One in a trillion-trillion-trillion chance?

Ok, lets take a breath here. What's this trillion-trillion and trillion-etc stuff?

(In my observation, economists say million, billion, trillion, and all their audiences hear is "big number." Innumeracy over scale has bad consequences when applied to public policy.)

One trillion is 1,000,000,000,000. (Yes, I'm using American billions, pretty much like everyone else now does.) This is written as $10^{12}$. A trillion trillion is $10^{12} \times 10^{12} = 10^{24}$ and a trillion-trillion-trillion is $10^{36}$, a one followed by thirty-six zeros.

To put that number in perspective, the age of the Earth is about $4.5$ billion years, or about $1.42 \times 10^{17}$ seconds. That's 142,000 trillion seconds. Note that this is much smaller than a trillion-trillion seconds (it's over one seven-millionth of a trillion-trillion), let alone a trillion-trillion-trillion. If you had seven million planets the same age as the Earth, and you picked at random one specific second in the history in one specific planet you'd would have about a one in a trillion-trillion chance of picking this precise second on this planet. A one in a trillion-trillion-trillion chance is one trillion times smaller than that.

So, something that has a one in a trillion-trillion-trillion chance of happening has to be a very low probability event. Shall we call anything less likely than that "almost impossible"? We shall.

So, here's how I make something almost impossible happen, over and over again, and you can too: shuffle a deck of cards.

Using only 52 cards (no jokers), there are $52! = 52\times 51 \times \ldots \times 2$ possible card shuffles, and $52! \approx 8.1 \times 10^{67}$. That number is $8.1\times 10^{31}$ times bigger than a trillion-trillion-trillion.

And yet, every card shuffle produces an event with 1-in-$8.1 \times 10^{67}$ probability. You and I can generate scores of these "almost impossible" events using a simple deck of cards.

(A little thinking will lead an attentive reader to the solution to this apparent paradox. It's not a paradox. I will post a solution here in a few days, if I remember :-)

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Just for fun, a simple brain teaser:

Imagine you have two decks of 52 cards (blue and red); what has more possible combinations, shuffling the two decks together and dividing into two piles of 52 cards by separating in the middle of the full shuffled two decks, or shuffling the two decks separately each into its pile?

(Yes, it's obvious for anyone conversant with combinatorics, but apparently not everyone is conversant with combinatorics. Common answer: "it's the same.")