I've been thinking about playing cards lately. Specifically, the odds of a specific shuffle configuration for the standard 52-card deck. This is actually a well-known thought experiment that professors of combinatorics and probability like to impress their freshman students with, but it’s not any less impressive because of it.

The idea is that there's an incomprehensibly huge number of ways you can shuffle a deck of cards. In combinatorics terms, there are 52! (factorial) possible shuffles. The gist of the factorial operation is to count how many ways you could order the items of a given set without allowing repetitions. For example, in our case, we have that for the first position in the deck you can pick any of the 52 cards, then for the second position you can pick any of the remaining 51 cards, then any of the 50 for the third position, 49 for the fourth, and so on. This eventually gives us a multiplication of all possible combinations for all positions. So, 52 * 51 * 50 * 49 * ... * 1, which is around 8 * 10^67, or, fully written out:

8,065,817,517,094,387,857,166,063,685,640,376,697,528,950,544,088,327,782,400,000,000,000

A big-ass number. For comparison, the estimated number of atoms in the observable universe is 10^82, which, yeah, it's a lot (a lot) bigger, but it's still mind-blowing that the number of possible shuffles of a common deck can even be compared to it.

To get the probability of getting a specific shuffle configuration, we do 1 / (8 * 10^67), which is crazy small. So small, in fact, that I would say it's almost impossible for two people to shuffle a deck in the exact same order. You could basically consider that every time you shuffle a deck of cards, you're creating a configuration that no human has ever seen before, and likely never will. Your very own, personal, unique shuffle.

...

I just did some searching to see if my numbers above were correct and found that there are a lot of people who have covered this in much greater detail than I ever could. One such person is czep, who proposes an awesome thought experiment to help us really appreciate just how big 52! is. I suggest you check out his post, though I'll write down the main points here so you can follow. The idea is to reflect on how much time it would take to count up to 8 * 10^67 seconds. It goes like this:

  1. Start a timer that will count down the number of seconds from 52! to 0.
  2. Stand on the equator. Walk around the world, taking one step every billion years.
  3. Every time you complete a lap (~40,075 km), remove one drop of water from the Pacific Ocean.
  4. When the Pacific is empty (~707.6 million km³), place a single sheet of paper on the ground.
  5. Refill the ocean. Repeat until the paper stack reaches the Sun (149.6 million km away).
  6. Check the 52! countdown timer. The first three digits haven’t even changed.
  7. Do this whole process a thousand more times. You're only a third of the way there.

It hurts my brain to even try to make sense of these time spans.

...

I should know better than to start looking up stuff online while I'm in the middle of writing a post since I always get sidetracked. Unsurprisingly, this is what happened today as well, and I saw that there are a lot of other cool examples of "unlikely" common things. Another one I thought was quite interesting was the likelihood of getting struck by lightning. According to weather.gov, the odds of being struck by lightning in your lifetime are 1 in 15,300. The odds of being struck twice in your lifetime are 1 in 234,090,000. And yet, even with this chance being so small, there are multiple cases of people struck more than once by lightning! Of course, it's probably a mistake to take these "probabilities" at face value since they're for the average human being, when in reality there are people who (either due to requirements of their life situation or because they simply lack common sense) are much more prone to being struck by lightning.

Another interesting probability I chanced upon was an approximation a guy named Dr. Ali Binazir made for how likely it is for you yourself to have been born. He approximates it at 1 in 10^2,685,000, which is just... no words. And yet, we're all here, aren't we? All here, taking everything for granted.

...

Seeing these numbers helps me appreciate just how full of such unlikely magic the world around us is. You're breathing; countless extremely unlikely configurations of molecules are constantly happening all through your body and all around you. Photons from the sun, which have themselves gone through mind-boggling journeys1, are bouncing on your skin and creating a cascade of incomprehensible proportions.

And yet... and yet... The news...

We're so caught up in ourselves that we're unable to see, to realize that we are not separate from everything, to realize we are everything.

Footnotes

  1. I would suggest you read up on the fascinating topic of radiative random walk, which tries to model how the energy produced inside the sun manages to escape. The estimate is that a given packet of energy produced by fusion can take ~100,000 years since its "creation" till it exits the "photosphere" and eventually travels to space. It's crazy to think that the light that warms our face might actually have originated way before the first human walked the Earth. ↩