Friday, October 31, 2014
Infrequently Asked Questions
Not exactly a stats post. Just a thinkin' about stuff and writin' about stuff post. These are some questions I've either been asked at some point, or that I ask myself every night as I fall asleep, fearful that the darkness has no answers, more fearful that it will speak answers I cannot bear to hear. Think of it as a Frequently Asked Questions post, using a liberal interpretation of the terms "frequently" and "asked."
Wednesday, October 1, 2014
Ten Card Draw
Sort of a sequel-post here, with more of the tarot math I was playing around with last time. Some of the references I used for that post discussed the idea that if you properly shuffle a deck of 52 playing cards, there's a decent chance that you've created a unique order of cards, never before seen in the history of shuffled decks. I wondered, what are the chances that any given tarot reading is unique in the history of tarot readings?
The spread pictured above is the Celtic cross spread, one of the more popular tarot spreads out there. Different people have different preferences for what each position signifies-- hell, there are only about three matches between the diagram above and the layout as I learned it-- but it's always ten cards, and the meaning of the spread depends upon the order the cards are drawn as well as the orientation in which they're laid down.
Given those parameters, we can calculate the total number of unique Celtic cross spreads:
Now we come to the second part of the question: what's the chance that, given how many possible Celtic cross spreads there are, no two Celtic cross spreads in the history of tarot readings have been identical? To answer this, we can use the math of the Birthday Paradox. If you haven't heard of it, the Birthday Paradox is the name given to the fact that you don't need as large a group of people as you might think before you start getting a pretty good chance that at least two people in that group share a birthday. If you have 23 people in a room together, there's about a fifty-fifty chance that there's a shared birthday among them. The linked explanation of the Birthday Paradox is better than any I could give, so I'll just let y'all educate yourselves there if you want to know more about it.
What's useful for our purposes is the shortcut formula near the end of that post: if you've got a pool of a given number of things, and you have an equal chance of drawing any one of the things, how many times do you need to draw from that pool before your chances of having drawn the same thing twice are about fifty-fifty? The precise math is complex, but we can get a decent estimate by taking the square root of the size of our pool-- or, a little more specifically, 1.177 times the square root of our pool. There are 365 possible birthdays out there (excluding leap years), and 1.177 times the square root of 365 is about 22.49, very close to the actual 23-person figure.
We have a pool of Celtic cross configurations of a known (if enormous) size. Roughly how many tarot readings would need to occur to reach an even chance of at least one repeat?
The answer: You'd need more than 80 billion Celtic cross tarot readings before the chances of a repeat reading reach fifty percent. Specifically, you'd need to do 80,482,750,652 tarot readings, or 11.3 for every human alive on Earth today. That is a big number.
I'm not exactly sure how to determine the total number of tarot readings that have ever occurred. If the average person has had a dozen Celtic cross readings in their lifetime, then we've probably reached 80 billion tarot readings total. But have they? I imagine the variance for that dataset is pretty big-- lots and lots of people who've never had a reading, versus enthusiasts who might have had hundreds. There's probably a way to estimate that, but it would take more effort than I'm willing to put in.
In any case, if every person on earth sat down and did ten consecutive Celtic cross spreads right now, there's a better-than-average chance that every single one of those readings would be unique. That's a staggering enough thought that I'm willing to say there's a good chance that your ten-card Celtic cross reading, while still bullshit, is your very own, never-before-seen, personal bullshit. And isn't that something?
The spread pictured above is the Celtic cross spread, one of the more popular tarot spreads out there. Different people have different preferences for what each position signifies-- hell, there are only about three matches between the diagram above and the layout as I learned it-- but it's always ten cards, and the meaning of the spread depends upon the order the cards are drawn as well as the orientation in which they're laid down.
Given those parameters, we can calculate the total number of unique Celtic cross spreads:
78 x 2 x 77 x 2 x 76 x 2 x 75 x 2 x 74 x 2 x 73 x 2 x 72 x 2 x 71 x 2 x 70 x 2 x 69 x 2 =
4,675,765,217,094,107,136,000
The point of all those 2s in there is to represent that each time a card is laid down, there are two possibilities for the way it faces. As you can see, there are a buttload of possible outcomes for a Celtic cross spread. Using standard US nomenclature for really big numbers, we can say that there are more than 4.6 sextillion possible Celtic cross spreads.
... the major motivation behind this blog post is that I calculated a number that gives me an excuse to say the word "sextillion." It is, scientifically speaking, the funniest number-related word.
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| A Saga-themed tarot deck would be AMAZING, btw. |
Now we come to the second part of the question: what's the chance that, given how many possible Celtic cross spreads there are, no two Celtic cross spreads in the history of tarot readings have been identical? To answer this, we can use the math of the Birthday Paradox. If you haven't heard of it, the Birthday Paradox is the name given to the fact that you don't need as large a group of people as you might think before you start getting a pretty good chance that at least two people in that group share a birthday. If you have 23 people in a room together, there's about a fifty-fifty chance that there's a shared birthday among them. The linked explanation of the Birthday Paradox is better than any I could give, so I'll just let y'all educate yourselves there if you want to know more about it.
What's useful for our purposes is the shortcut formula near the end of that post: if you've got a pool of a given number of things, and you have an equal chance of drawing any one of the things, how many times do you need to draw from that pool before your chances of having drawn the same thing twice are about fifty-fifty? The precise math is complex, but we can get a decent estimate by taking the square root of the size of our pool-- or, a little more specifically, 1.177 times the square root of our pool. There are 365 possible birthdays out there (excluding leap years), and 1.177 times the square root of 365 is about 22.49, very close to the actual 23-person figure.
We have a pool of Celtic cross configurations of a known (if enormous) size. Roughly how many tarot readings would need to occur to reach an even chance of at least one repeat?
The answer: You'd need more than 80 billion Celtic cross tarot readings before the chances of a repeat reading reach fifty percent. Specifically, you'd need to do 80,482,750,652 tarot readings, or 11.3 for every human alive on Earth today. That is a big number.
I'm not exactly sure how to determine the total number of tarot readings that have ever occurred. If the average person has had a dozen Celtic cross readings in their lifetime, then we've probably reached 80 billion tarot readings total. But have they? I imagine the variance for that dataset is pretty big-- lots and lots of people who've never had a reading, versus enthusiasts who might have had hundreds. There's probably a way to estimate that, but it would take more effort than I'm willing to put in.
In any case, if every person on earth sat down and did ten consecutive Celtic cross spreads right now, there's a better-than-average chance that every single one of those readings would be unique. That's a staggering enough thought that I'm willing to say there's a good chance that your ten-card Celtic cross reading, while still bullshit, is your very own, never-before-seen, personal bullshit. And isn't that something?
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