Recurring Card Patterns and Probabilities

We previously looked at the standard deviations for sets such as reversals. We compiled a table to provide distribution ranges in order to provide a good idea of the counts that might indicate when too many or not enough of any particular pattern appeard in a spread. This post expands upon the previous, and gives you a tool to answer questions about the odds of this or that pattern of cards appearing in a spread.

If you’re like me, you have no doubt looked at what seems to be a wildly notable pattern of cards having something dramatically in common in a spread, and wondered, "what’s the odds of that?" We’ll start with a simple example. In a three card spread, what are the odds of getting three Eights? I’ll tell you. The odds are 19,019 to 1.

Do such odds matter, other than as a curiousity? It sort of sounds like an old Star Trek episode in which Mr. Spock tells Captain Kirk:

Captain, there are approximately one hundred of us engaged in this search, against one creature. The odds against you and I both being killed are 2,228.7 to 1.

But actually, as far as tarot is concerned, yes, the odds may indeed matter. At a simple universe-hits-you-smack-between-the-eyes level, if you see a spread with odds of nineteen thousand to one against, you’re going to pay attention. You’re going to tell the querent they better listen up.

And in fact, the 19th and early 20th century experts took note of these patterns. You’ll find that Waite, in The Pictorial Key to the Tarot, has an entire section on "The Recurrence of Cards in Dealing." He provides additional meanings for these patterns, which we’ll list at the bottom of this post.

In the case of those three upright Eights, Waite writes that it signifies marriage. He no doubt meant these meanings in the context of a Celtic Cross. With more cards, probability increases, and the odds drop. In a layout of ten cards, the odds for the three Eights drop all the way down to 175 to 1. This is probably a good thing, because querents sometimes like to be told a marriage is in the cards.

One important note: Waite’s additional meanings only apply if they are all upright or all reversed. In fact, the upright patterns have one set of additional meanings, and the reversed have another.

By the way… the odds of four Eights (or all four of any other number or a particular court card) in a ten card spread? 6793 to 1. With the spreadsheet, which you can download at the link below, you can ask (and answer) your own questions.

Tarot Odds Calculator

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Download Spreadsheet

Gimme Some Spock

First though we’ll explain the general mechanics of it. The heart of it is a hyper-geometric function. It basically just figures out the probability of the first element in a series being drawn as one element from the universe of choices. It then reduces the universe by one, gets the probability of a second element from this reduced universe, then calculates the probability of both element one and element two happening together. It repeats for as many elements we have to calculate. The result is a percentage, and the conversion to odds is just a bit of juggling fractions.

Mind you, were it not for the fact that most modern spreadsheets include hypergeometric funtions, this would be a great deal, a very great deal of work. I should mention I never took a course in statistics, but given that I’m reasonably good in math, that I seem to follow directions well, and that I know my way around spreadsheets and such, I think I can say that the formulae are pretty trustworthy. The spreadsheet includes some rounding; so it’s possible a more precise statement of, for example, the last odds quoted might actually be 13,587 to 2, but at the degrees of numbers we’re looking at, I don’t think it matters. If you do… the spreadsheet is yours to modify, have at it. Just let me know about it if you improve it or find a fix necessary.

You can confirm the reliability fairly easy. Take the odds of one card of 78 in a one card spread: the spreadsheet will show you 78 to 1. Now leave it at one card in a ten card layout. The odds are 8 to 1.

To use the spreadsheet, we need to know four things in order to ask what the odds are:

• First is the quantity of the type of thing you want to appear (or which did appear). So, in the case of the three Eights, three.
• Next is the how many in the spread; in other words, if we want to know the odds of three Eights out of ten cards, we would make ten the parameter.
• Then we need to specify how many of the type of that thing exist. In this case, there exist four Eights in all, so we would specify four.
• Finally we need to specify how big the universe of choices is: in this case, seventy eight, the number of cards in the deck.

Above we see an example. We asked, what are the odds of all four of the same number (or court title) appearing in a ten card spread? We enter 4 for the number of the type of element we want, 10 for the number of cards in the spread, 4, for the total that exist within the deck of that element, anf finally, 78 for the deck. The answer appears the bottom cell: 6793 to 1.

Let’s take a stratospheric example or two:

In the example above, we ask what are the odds of having ten major arcana in a Celtic Cross? We entered ten for the type, ten for the spread, 21 because there are 21 major arcana (excluding the Fool), and 78 for the deck. The odds are 3.5 million to one! But that’s just any ten majors. If we picked out exactly ten cards from the majors (or any ten cards of any type) and specified we wanted to know the odds that those ten, and only those ten would appear, we’d change the 21 in the third box to 10… and we find the odds are 1,258,315,963,905 to 1.

Such high odds sort of highlight the small miracle that any particular spread of drawn tarot cards represents, doesn’t it? The universe deals you exactly the cards you see in front of you at any given time. They are yours, and no one else’s.

So far, though, we’ve only talked of probabilities and quantities, and only touched upon meanings when we mentioned Waite’s take on it. But there’s more.

Take the astrological or elemental influences (no doubt you’ve already realized that the four-of-a-kind odds cover the qabalistic influences in the minors!).

But we know that each planet (except Mars) rules over six cards (Mars over seven). We know that each element (except Earth) has influence over fifteen cards (Earth over fourteen). Hopefully, Pamela Coleman Smith’s illustrations will alert us visually when, say, a preponderance of Mercury appears in a spread. But it’s all together different to feel surprise at six cards influenced by Mercury appearing in a ten card spread, versus realizing the hard fact that they represent 1.2 million to one odds against that having happened. (Buy penny stocks if that ever happens… Mercury is the god of business and commerce!)

Getting six "planet cards" would no doubt be an extreme case, but it illustrates that knowing some of these odds may help you consider the importance of card patterns in a reading.

That should be the takeaway: the importance is in the patterns, and if the drama of the numeric probabilities drives that home, then this exercise has been valuable.

Final note: You’ll note that the spreadsheet doesn’t take reversals into account. Presumably, multiplying the 78 by 2 will account for reversals; but frankly, I haven’t given it that much thought.