# Custer’s Last Stand

By Bobby Neal Winters

We are told that a prophet is not without honor except in his own home town. And it’s true that those of us who know us best have the most trouble believing that we might know anything ourselves. This is especially true, in my opinion, when you practice mathematics. I think this is because that it can be so technical that non-professionals can’t tell what is relatively easy from what is hard because they don’t understand either.

At times it has been important to me to convince my kinfolks that I do know something about mathematics. This is made difficult for me because I am not very good with arithmetic. If you ask me to add up a column of figures, I do like everyone else and reach for a calculator. Add to this the fact that my expertise is in the topology of non-compact three-manifolds which is something that doesn’t just naturally come up in conversation after Thanksgiving dinner.

I have learned a tiny bit of statistics by teaching it over the years and that has provided an opportunity to help at least one of my kinfolks recently. That kinsman was my cousin Custer.

Custer is the grandson of my Aunt Vidalia. He was her special pet when he was growing up and she spoiled him rotten. His parents spoiled him rotten as well. They are fun to spoil, but sooner or later there is a price to pay, and Custer has had a problem that in some circles is referred to as failure to launch.

That is to say that they can’t get him to move out of the house; he just hangs around and obsesses on Bigfoot all day.

The reason for this has to do with work. It’s not that he can’t get a job. He’s had lots of jobs. Vidalia has lots of friends who have either given him jobs or found jobs for him. It’s keeping the job that’s the problem. Some of the elders in the family have been of the opinion that if he spent half as much time working as Vidalia’s friends have in finding work for him that he would be president of Standard Oil by now, but I believe in giving everyone a chance.

His latest job had been at the Sucker-Rod Museum. According to its brochure, it was the home of the world’s largest pile of sucker-rods. For those of you who didn’t grow up in the oil field, a sucker-rod is a cylindrical piece of metal. They are about an inch in diameter and about 30 feet long. None of this is important to what comes next, but I didn’t want you to be worried about not knowing that.

The job had been given to him by one of Vidalia’s friends who is big in the oil field. He’d set up the Sucker-Rod Museum and had needed someone sit at the reception desk, invite the visitors to sign the book, and to keep track of the number of folks who visited. Previously this job had been given to a retired old-field worker who was confined to a wheel chair. The job had opened up when the old fellow had died at the age of 92.

The fellow had taken the job seriously and had kept an immaculate set of records. I know this because I’d seen them as a part of a project I’d done for Custer.

Not long after getting the job, Custer had e-mailed me with a project he was working on. He had the records that his predecessor had collected and wondered if there were ways of predicting the number of visitors to the Museum on a given day.

Now, I get nervous when people start using that word “predict.” We can make statements about trends over time given certain assumptions, but we can’t actually be prophets. People want magic, however, and sometimes it’s hard not to get sucked in.

I answered his e-mail and told him to send me a copy of his records and I would see what I could do.

The Museum is opened 6 days a week, Tuesday through Sunday, 52 weeks a year. It’s closed on Christmas and Easter, so that usually works out to being open 310 days a year. The last year that I had full data on, it had 300 visitors. This was an average of 0.97 visitors a day. The other years were off from this slightly but not by much.

The idea of having an average of 0.97 visitors can be somewhat comical, as we try to imagine what 0.97 of a person is. Would these be old oil-field workers missing a few fingers or a hand? Maybe that’s not so comical. Anyway, that is not how averages work, of course. That this means is that some days several people would come in and some days nobody would be there. While on any given day, it is not necessarily predictably what will happen, one can sometimes say something about the distribution. By this I mean to say that there will be a certain fraction of the days when no one will come in, there will be a certain fraction of the days with one person will come in, a certain fraction when two will come in, and so forth.

Often situations like this will follow the Poisson Distribution, which is also known as the Law of Rare Events. This is kind of misleading because the Law of Rare Events can also model very busy places like restaurants and barbershops. The word rare is a reference to on any given day the probably of a particular person going to a particular restaurant might be small. The distribution depends upon the average number of visits per day, such as the 0.97 for the Sucker-Rod Museum.

I examined all the data he sent me, and it turned out that the pattern fit the Poisson Distribution very nicely. I sent him a report saying that on 38 percent of the days there would be no visitors, on 37 percent of the days there would be one visitor, on eighteen percent there would be two visitors, on six percent there would be three, and on one percent there would be four.

I swear that is all that I told him.

A few weeks later I got a hot call from Custer’s father, in whose basement Custer is living, telling me that I’d gotten Custer fired.

“What?” I asked, hurt and confused.

“He said that you’d done some math and he used it and was fired because he used it.”

After saying that I didn’t see how what I told him could get him fired, I said that I was sorry that he’d lost his job, then I hung up, and made a few phone calls. After talking to several people, the following picture emerged.

Custer observed that 93 percent of the days there would be two or fewer visitors. He then deduced—correctly—that only on seven percent of the days were there more than visitors. He then developed the habit of closing the museum after the second visitor left deducing—incorrectly—that where was only a seven percent chance that anyone else would come.

Over the course of time there were complaints from people who showed up to the museum wanting to see the world’s largest pile of sucker-rods only to find the museum closed.

My final call was to my Aunt Vidalia. Who explained how it came to an end.

“Custer has always been a big fan of Sasquatch. It all came crashing down on the weekend of the Bigfoot Festival in Honobia, Oklahoma,” she pronounced Honobia as Hoe-nubby just as its inhabitants do. “He’d taken that weekend off without telling his boss. He was afraid he wouldn’t get paid if he told him. Then the boss came by. Well, he just didn’t understand.”

I commiserated a bit with Vidalia over this before I said good-bye.

Custer made quite a few mistakes here. The non-mathematical ones I leave to you. The mathematical ones are a bit subtle and don’t have anything to do with the Poisson Distribution which I’d done correctly.

Custer’s problem was his assumption that if he left after the third visitor there was only a seven percent chance that there would be another one. It turns out that if you’ve already had three visitors, then there is about a thirty percent chance of having more visitors. This is because already knowing that you have had three visitors changes the initial assumptions of the problem.

It’s called conditional probability and it catches a lot of people by surprise. The one thing that is not a surprise is that Custer is out of work, and the family is blaming it on me.

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