$7.11 at 7:11 PM on 7/11 at 7/11: What Are The Chances?

Let’s take something silly and turn it into something shiny. And, in doing so, relearn the invaluable lesson that nothing has a probability.

Somebody posted this receipt from a sale at a 7/11 in Jacksonville, Florida, and I was asked to comment on it. I’m only showing the meat of the receipt here, but the fellow who owned it had the whole thing framed because of its peculiar coincidence of numbers.

As you can see, the receipt was for $7.11, made on 7/11 at 7:11 PM, and at a 7/11. The poster said “The odds of this happening are like 1 in 525M+”.

Is he right?

What is the chance this happened? One. The probability is 1; it is certain, because, on the assumption the evidence before is genuine, it happened. The probability is always 1 in all such cases where we believe the evidence true.

This right here should be proof enough that things, events, don’t have probability. You, for instance, do not have the probability of dying by being flattened by a bus, or from any malady. All probability is conditional on the evidence you assume. There are no exceptions.

Still, this “1” is not a satisfying answer here. It does not capture the surprise or curiosity of this coincidence. And anyway, if the probability of all things is “1”, there would seem to be no need for probability to exist as any kind of intellectual endeavor.

Which means we’re asking not “What is the probability this happened?” in these kinds of questions, but “What is the probability things like this, but not this, happened?” And that answer depends on what you assume, you knowledge or supposition about the thing and its circumstances.

Another way of phrasing it is in asking “How unusual is this kind of thing?” And that depends entirely on what you mean by the thing.

Now the analysis which follows is silly, mere fluff, and it would not be worth doing at all, except that the techniques to answer it are the same in any instance of questions like these. For instance, from John Dunn I received word about a “cancer cluster” in children in Lareda Ranch, California. Somebody thought the number of kids being diagnosed with a certain form of cancer was too unusual to be “random”. The person writing about it said “No matter the cause, the Ladera Ranch pediatric cancer cluster appears statistically unfathomable.” Statistically unfathomable!

Since nothing is random, but our knowledge of things if often uncertain, the real question here is what caused the cancers. By all the misuse and exaggeration of probability language, the writer of that article meant to imply it must have they cause he had in mind (pesticides). Whether that is true, I have no idea, but saying something is “statistically unfathomable”, or in less fantastics language “improbable”, is no proof the cause is true.

This is backward thinking; it inverts the question and provides a false answer. The same thinking that gives us P-values. What we want to know is, given the evidence we assume, what is the probability the cause we put forward is true. To say that because a thing is “statistically unfathomable” the cause must therefore be true is always a fallacious argument.

Since looking into the cancer cluster is too hard, given we don’t have any of the data, especially data on the diagnoses process, itself ever important, we’ll instead show how analyses are done with 7/11 sales, where we have a better idea of the causes of sales.

First thing to note is that we’re dealing only with 7/11 stores. Not Holidays, or WaWas, or any other kind of convenience store. After all, the sale would never had been noticed unless it were at a 7/11. This is important because we have to be diligent and detailed in defining what the thing is in which we take an interest.

Second is to define the event itself. Sales with exactly $7.11, those only on 7/11, and at 7:11 PM or, one would suppose, 7:11 AM. That is an assumption I make. If you agree with it, then you’ll agree with my calculations (up to that point, anyway), and if you don’t agree, then you will come to a different probability. Because you assume different evidence. Thus the debate in all probability questions, once the thing is decided upon, is about the evidence assumed.

The rest of my evidence is surely up for debate. I looked up some statistics about sales in convenience stores, but I cannot vouch for any of them. I don’t know what I don’t know about 7/11 sales, and I have little personal experience, since the nearest one to me is an hour away and I haven’t been to it in over a decade. I’ll give you my sources and ideas, but again these are not that important, but keep in mind the method, because it would be the same for those purported cancer clusters.

We could get transactions of $7.11 any day of the year, and only two possibilities would make it a coincidence: July 11th, or 7 November, in the two common ways of writing dates that can get us “7/11”. Of course, only one of these date formats is used by the company, and we know this because the way the receipt is printed. So it’s just July 11th that interests us.

What if the $7.11 transaction were on 3 February? Would that also be a coincidence? Maybe, since it was still at a 7/11 (we assume) and at the right time, but one different in nature, and seemingly not as rare.

From various industry sources (here, here, and here) we learn things like the average non-fuel-related transaction is $12.78, and maybe 90% or so are from $1 to $20, with a only 10% higher amounts. Sales can only be the nearest penny, so the possibilities are $1.00, $1.01, and so on. Individual store sales vary, but the number of transactions average about 1,450. Seven-Elevens are mostly open 24 hours, with the bulk of the sales coming sometime in the early afternoon. If Google business indicators are anything to go on, this picture of activity from the store closest to me is a useful guide to traffic.

There are also, as of the latest number I saw, some 13,427 7/11 stores in the States. And many others elsewhere in the world.

Given our sources, perhaps something like this, a discretized gamma distribution (with parameters 4.3 and 0.34) is reasonable for the transaction price probabilities:

This only looks continuous; in fact, it’s the probability for each possible transaction, from $1.00 to $40.00, each penny intervals. The probability for $7.11, conditional on this evidence, is about 0.0006. Which is a bit less than 1 in a 1,000.

Using the picture of transaction times, maybe something like this, the probability for each minute of the day, using a discretized wrapped normal (with central parameter of 990 minutes and spread of 600 minutes):

This doesn’t look exactly like observed time distribution, because the axis has shifted, starting here at midnight (o minutes). The peak is at 4:30 PM, as in the data. The difference in peak probability to minimum probability is about 5 times, which also looks like the data. The two vertical lines are at 7:11 AM and 7:11 PM. Notice the probabilities only look like huge swings, but the differences are small.

The 7:11 AM gives a probability of about 0.0003, and the 7:11 PM is about 0.001. Totaled, for AM or PM, we get 0.0014.

What about the date? Well, at first we are assuming we’re only looking at the specific date 7/11, else there never would have been a coincidence to notice.

We have solved this problem: Given standard industry statistics, and their approximations as noted, what is the chance of a transaction $7.11 at either 7:11 AM or 7:11 PM? We multiply the two probabilities we just estimated:

price_ p x time_p = 8.8 x 10^-7.

That’s almost 1 in a million (which would read 10 x 10^-7 = 1 x 10^-6).

But that’s not the answer to our main question yet. For one thing, there can be more than one transaction in a minute. The number of transactions per minute will be the same shape as the times of transaction. If each store has 1,450 transaction a day (as our sources suggest), then we’d multiply the time probability by this to get an average value for the number of transactions, which is just about 2. (The slowest parts of the day get about a half a transaction a minute, and the busiest about 1.75 transactions per minute, on average.) Without going into detail it means accounting for the possibility of more than one sale a minute won’t adding anything important for us.

But adding numbers of stores would. If we account for all 13,427 stores, the chance that at least one of these stores sees a sale of $7.11 at 7:11 AM or PM, assuming we’re looking at 7/11 and the other information we gathered, is 0.012. Just over a 1% chance. Not anywhere close to 1 in 525 Million.

That’s for one year’s 7/11. Over 10 years the chance is 11%.

Of course, this is all order-of-magnitude, because over a decade numbers of stores will change, and with inflation surging the chance of $7.11 in sales changes, and so on.

Call it one in ten, then. Not large, but not so small, neither. That’s before we account for somebody noticing the coincidence, which is an entirely different kind of event.

But you get the idea. You can go on and on endlessly bickering over the evidence, even for simple problems like this. Imagine how much more difficult it is for supposed cancer clusters.

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