# How Not To Count Fish

I’d discussed a probability/sampling-based method to estimate the number of fish that might be in our pond out back, and then some of the errors that have to be handled if you want to have a reliable result. Now, I want to get into why the method doesn’t work, at least not without much greater insight into goldfish behavior than simply catching a couple and releasing them will do.

Catching a sample, re-releasing it, and counting how many of that sample we re-catch later on is a logically valid method, provided certain assumptions the method requires are accurately — or at least accurately enough — close to the way the actual thing works. Here are some of the ways goldfish fall short of the ideal.

First faulty assumption: Goldfish are perfectly identical. In this goldfish-trapped we make the assumption that there is some, fixed, constant probability of a goldfish being caught in the net. We have to assume that this is the same number for every goldfish, and that it doesn’t change as goldfish go through the experience of getting caught and then released. But goldfish have personality, as you learn if you have a bunch in a nice setting and do things like try feeding them koi treats or introduce something new like a wire-mesh trap to their environment. Some are adventurous and will explore the unfamiliar thing; some are shy and will let everyone else go first and then maybe not bother going at all. I empathize with both positions.

If there are enough goldfish, the variation between personalities is probably not going to matter much. There’ll be some that are easy to catch, and they’ll probably be roughly as common as the ones who can’t be coaxed into the trap at all. It won’t be exactly balanced unless we’re very lucky, but this would probably only throw off our calculations a little bit.

Whether the goldfish learn, and become more, or less, likely to be trapped in time is harder. Goldfish do learn, certainly, although it’s not obvious to me that the trapping and releasing experience would be one they draw much of a lesson from. It’s only a little inconvenience, really, and not at all harmful; what should they learn? Other than that there’s maybe an easy bit of food to be had here so why not go in? So this might change their behavior and it’s hard to predict how.

(I note that animal capture studies get quite frustrated when the animals start working out how to game the folks studying them. Bil Gilbert’s early-70s study of coatis — Latin American raccoons, written up in the lovely popularization Chulo: A Year Among The Coatimundis — was plagued by some coatis who figured out going into the trap was an easy, safe meal they’d be released from without harm, and wouldn’t go back about their business and leave room for other specimens.)

Second faulty assumption: Goldfish are not perfectly identical. This is the biggest challenge to counting goldfish population by re-catching a sample of them. How do you know if you caught a goldfish before? When they grow to adulthood, it’s not so bad, since they grow fairly distinctive patterns of orange and white and black and such, and they’ll usually settle into different sizes. (That said, we do have two adult fish who were very distinct when we first got them, but who’ve grown into near-twins.)

But baby goldfish? They’re basically all tiny black things, meant to hide into the mud at the bottom of ponds and rivers — their preferred habitat — and pretty near indistinguishable. As they get larger they get distinguishable, a bit, and start to grow patterns, but for the vast number of baby fish there’s just no telling one from another.

When we were trying to work out whether some mice we found in the house were ones we had previously caught and put out in the garage, we were able to mark them by squiring some food dye at their heads as they were released. The mice would rub the food dye from their heads onto their whole bodies and it would take a while before the dye would completely fade out. (We didn’t re-catch any mice, although it’s hard to dye a wild mouse efficiently because they will take off like bullets. Also one time when we thought we’d captured one there were actually three in the humane trap and you try squiring the food dye bottle at two more mice than you thought were there, fleeing.) But you can see how the food dye wouldn’t work here. Animal researchers with a budget might go on to attach collars or somehow otherwise mark animals, but if there’s a way to mark and track goldfish with ordinary household items I can’t think of it.

(No, we will not be taking the bits of americium in our smoke detectors out and injecting them into trapped goldfish; among the objections, I don’t have a radioactivity detector.)

Third faulty assumption: Goldfish are independent entities. The first two faulty assumptions are ones that could be kind of worked around. If there’s enough goldfish then the distribution of how likely any one is to get caught will probably be near enough normal that we can pretend there’s an identical chance of catching each, and if we really thought about it we could probably find some way of marking goldfish to tell if we re-caught any. Independence, though; this is the point on which so many probability-based schemes fall.

Independence, in the language of probability, is the principle that one thing’s happening does not affect the likelihood of another thing happening. For our problem, it’s the assumption that one goldfish being caught does not make it any more or less likely that another goldfish will be caught. We like independence, in studying probability. It makes so many problems easier to study, or even possible to study, and it often seems like a reasonable supposition.

A good number of interesting scientific discoveries amount to finding evidence that two things are not actually independent, and that one thing happening makes it more (or less) likely the other will. Sometimes these turn out to be vapor — there was a 19th-century notion suggesting a link between sunspot activity and economic depressions (because sunspots correlate to solar activity, which could affect agriculture, and up to 1893 the economy and agriculture were pretty much the same thing) — but when there is a link the results can be profound, as see the smoking-and-cancer link, or for something promising but still (to my understanding) under debate, the link between leaded gasoline and crime rates.

How this applies to the goldfish population problem, though, is that goldfish are social creatures. They school, loosely, forming and re-forming groups, and would much rather be around another goldfish than not. Even as babies they form these adorable tiny little schools; that may be in the hopes that someone else will get eaten by a bigger fish, but they keep hanging around other fish their own size through their whole lives. If there’s a goldfish inside the trap, it is hard to believe that other goldfish are not going to follow it just to be with the company.

Indeed, the first day we set out the trap for the winter, we pulled in all but one of the adult fish, all of whom apparently followed the others into the enclosure. I’m sorry I couldn’t photograph that because it was both adorable and funny to see so many fish just station-keeping beside one another — they were even all looking in the same direction — and waiting for whatever might happen next. Throughout the months we were able to spend bringing in fish, the best bait we could find was to have one fish already in the trap, and a couple days we did leave one fish in a few more hours or another night so that it would be joined by several companions the next time we checked.

So that’s something which foils the catch and re-catch scheme: goldfish are not independent entities. They’re happy to follow one another into trap. I would think the catch and re-catch scheme should be salvageable, if it were adapted to the way goldfish actually behave. But that requires a mathematician admitting that he can’t just blunder into a field with an obvious, simple scheme to solve a problem, and instead requires the specialized knowledge and experience of people who are experts in the field, and that of course can’t be done. (For example, I don’t actually know that goldfish behavior is sufficiently non-independent as to make an important difference in a population estimate of this kind. But someone who knew goldfish or carp well could tell me, or tell me how to find out.)

For those curious how the goldfish worked out, though, we were able to spend about two and a half months catching fish before the pond froze over for the winter, though the number we caught each week dropped off as the temperature dropped. We have them floating about in a stock tank in the basement, waiting for the coming of spring and the time the pond will be warm enough for them to re-occupy it. We also know that at least some of the goldfish we didn’t catch made it to, well, about a month ago. I’d seen one of the five orange baby fish who refused to go into the trap through a hole in the ice then. It was holding close to the bottom but seemed to be in good shape.

This coming year should be an exciting one for our fish population.

## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

## 14 thoughts on “How Not To Count Fish”

1. AR says:

I’m interested in this idea of statistical independence. By this measure, people are not very independent either!

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1. Well, it depends on the frame, really. In some ways people can be treated as statistically independent entities: all the people in a mall food court, for example, could be modeled as equally likely as every other person to go to the McDonald’s, the sandwich shop, the Sbarro’s, or the mediterranean food grill, at least for the purpose of figuring out pedestrian traffic flows, or how tables would be best arranged, or matters like that.

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1. AR says:

Is that because there are so many people going through a mall, making group decision-making and following statistically unimportant?

I think I see dependence sometimes on facebook or Amazon reviews… whoever makes the first comment on a post or leaves the first review on a movie usually sets the tone for the majority of the other reviews and comments. It seems like social media has revealed people’s overall lack of individuality and their tendency to follow. (Blogging is, of course, the shining exception – the online bastion of originality.)

So for instance, I wonder what the outcome would be if one of those restaurants got models to walk in and out of their doors all day, and sit at outside tables laughing and eating and raving about the food.

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1. Well, independence in this context just means that the probability that (say) this party is going to that food stall doesn’t depend on what any other parties are doing. That is, it doesn’t matter if the last four parties entering all went to the hot dog place; that doesn’t make the next party to enter more — or less — likely to go to the pizza place.

(There are limits on this, of course. If there’s a line of 200 people at the hot dog place and nobody anywhere else, I would skip my hot dog plans. Or suppose that the hot dogs have got to be too good to pass up, I guess.)

The sort of dependence you’re describing seems to be more of an anchoring effect, one of the strange and creepy aspects of human decision-making. If asked to give their opinion on something, people will tend to give an answer that’s close to whatever the last thing they heard was, even if it had nothing to do with whatever they’re supposed to decide. The tone-setting effect of the first comments is probably a reflection of that effect in less-quantifiable matters.

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2. This is so interesting Joseph… I agree with the commenter above as I found that the excerpt called faulty assumption: Goldfish are independent entities, were outstanding.
Thanks for sharing this information and problems on statistics. Best wishes :star: Aquileana :D

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3. I’m going to read together with my 9 year old son who loves fish and real life math. Thanks, Joseph :D

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1. Aw, good. Hope you enjoy.

We’re hoping the weather will be warm enough in the next month that we can transfer the fish back into the pond. I can think of a mathematical problem that results from this, but it’s a less obviously appealing one than simply “estimate the pond’s fish population”.

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1. Have I mentioned my son-in-law who is a grad student in physics? Once he gets through prelims this spring (he’s hyper studying), I will send him your blog. I think he’ll really like it.

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1. I don’t remember that you had. That’s neat and I do hope he enjoys, although I also remember keenly how much work went into prelims. Analysis particularly was a hard one for me to get through.

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1. I love my son-in-law has a son. I met him when he was 19 and contemplating going to college for math. He’s first generation college and I couldn’t be more proud of him. We are close and I’m feeling his prelim pain!

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1. Oh, that’s great for him. I hope the prelims go well. Grad school was wonderful at least for me, and my love, although it is a lot of very particular challenges, many of them more of endurance and tolerance for the grad school lifestyle than anything else. I was able to come out the far end successfully, but it was a close-run thing at points, with prelims one of those points.

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