Jeremy recently passed along to me a conjecture that he’d heard:

>if you shuffle a deck of cards well, then it’s likely that never in history has a deck of cards been in that exact order.

I mentioned this to Sue, Michael and Megan in Portland, and there was general disbelief. I hadn’t done the math, but was inclined to believe it, because I knew 52 factorial was a really big number. Well, here’s the math, I believe the conjecture.

First, a quick intro to card combinatorics: If I have a deck with one card in it, how many orderings of that deck are there? Answer: 1. Two cards? Two orderings (one of them comes first, the other one second). Three cards? Six orderings. There are three cards the first one could be. After that, there are two ways the remaining two cards could be ordered, as discussed in the preceding sentence. Four cards? 4 * 3 * 2 * 1 = 24. Etc. That practice of multiplying each whole number less than or equal to a number is called *factorial*, and is represented in our math system by an exclamation point. So, for five cards, we have 5! = 5 * 4 * 3 * 2 * 1 = 120 orderings.

52! = 52 * 51 * 50 * … * 2 * 1 = 8 * 10^67

10 ^ 67 is a huge number. Consider:

there are about 6.5 billion people alive today. I’ve heard it said that one out of every 10 people is presently alive. That means that fewer than 100 billion people have ever lived. That’s 10 ^ 11.

Say that each of those 10^11 people had been shuffling decks of cards at a rate of a thousand different orderings every second for one million years each. We have:

10^11 people * 10^6 years * 365 days/year * 24 hours/day * 60 min/hour * 60 sec/min * 10^3 orderings per person per second = 3.2 * 10^27

10^27 is huge, but it’s much smaller than 10^67. Don’t think “it’s about 1/3 the size”. No, it’s the difference between 1 with 27 zeros and 1 with 67 zeros. That’s about a 1 in 10^40 chance. It is very unlikely that a well shuffled deck of cards is in the same order as any other well shuffled deck of cards has ever been. You’re more likely to be dealt a natural royal straight flush in five cards in 4 consecutive games. In fact, that’s an understatement.