I like to run. And unlike most Americans, I don’t normally stop to pick up pennies when I spot them. But to add some variety to my runs, I decided to run an experiment. For the month of May, I collected all the coins I saw.

The data

In May, I ran every day, but on two of those I was out of town for a race, so I excluded those from the dataset. Each day, I recorded my mileage (measured by GPS) and the coins I collected. All of the data is available in csv format over at my GitHub. There’s extra information there like my pace and whether I ran a loop or an out-and-back, but with only a month of data, I don’t think I can do any reliable analysis on these factors.

The money

Over the course of the month, I collected:

18 pennies

2 nickels

6 dimes

1 dog wash token

totaling $0.88. If I had a dog, the token might be worth more to me than all the other change combined, but it’s a weird enough outlier that I’ve excluded it from the rest of the analysis.

The easy math

I ran 128.3 miles in total. So that works out to $0.007 per mile. On average, I found one coin (of any denomination) for every 5 miles I ran.

For comparison, a professional runner making $70,000 per year and running something like 130 miles per week is making over $10 per mile.

Naïve variance

Looking at the month, there was a lot of variance from day to day. You can see most days I found nothing, and a handful of good days accounted for most of my coins. So even though I made 7/10 of a cent per mile, I can’t guarantee that if I run 3 miles, I’ll find a penny or two.

The amount of money I found on each day, normalized for distance.

I can try to get a handle on how much variance there is by treating each run as an independent measurement of the dollars-per-mile rate and computing the standard deviation. When I do this, I get that I can expect my running income to vary by plus or minus a penny per mile.

Assuming the dollar-per-mile rate is normally distributed predicts I’ll be losing money on some runs.

But that’s not a satisfying answer. I don’t lose money running. The mistake I’ve made here is assuming my data is normally (Gaussian) distributed. The normal distribution is symmetric around the mean, and has tails that go off to plus and minus infinity. Under this model, I’d lose money almost a third of the time. Fortunately, my actual money collection rate has a hard floor at zero.

A better model

In my physics career, I thought a lot about radioactive decay. And I can make an analogy between that and my coin collecting:

Decays either happen or they don’t; there are a whole number of them in the interval. Similarly, I either find coins or I don’t. There are no fractional coins.

Even though I can’t predict individual radioactive decays, I can predict how many will happen in an interval of time. Likewise, I can’t know when I’ll find a coin, but I’m trying to predict how many I’ll find in an interval of distance.

The average rate of radioactive decays doesn’t change over time (at least not significantly if we’re talking about relatively short intervals compared to the half life). Similarly, I don’t expect that the amount of change dropped will vary too much from day to day or block to block.¹

Radioactive decays are independent of each other. When something decays, it doesn’t make the next decay any more or less likely to happen in the future. Likewise, I expect that if I find one coin, that’s not going to make it more or less likely to find one further along.²

In a system that has the properties I’ve just mentioned, the Poisson distribution provides predictions for how many events, like radioactive decays or found coins, happen in an interval.

If I call the “coins per mile” rate λ, and the distance I run d, then the probability I find n coins is:

For the Poisson distribution, the expected mean number of coins will (λd), so I can estimate λ by dividing the total number of coins I found by the total distance. The best estimate is that I find 0.20 coins per mile³.

What can I expect?

Let’s say I go out for a typical-length run of around 4 miles. Plugging the value I got for λ in, I can expect to find:

0 coins 44% of the time

1 coin 36% of the time

2 coins 15% of the time

more than 2 coins 5% of the time

Probability of finding n coins on a 4 mile run

This seems at least plausible. I did have a lot of days when I found no coins, and a few rare days when I found many coins.

Quantitatively, I had more days (around 80%) with zero coins than this predicts (around 45%). If I look at the uncertainty on my estimate of λ, which I’ll take as the 95% confidence interval, a smaller λ of 0.13 coins per mile is still plausible. This would give closer to 60% of days with zero coins, still lower than what I got.

It could simply be that with only 12 runs around 4 miles, I got unlucky. It could also be evidence my model isn’t a great description of reality. I’ll need to collect more data to find out either way.

If I estimate the number of coins I expect to find on the roughly 2000 miles I plan to run this year, and consider the value of the “average” coin I find, I’d expect to make about $12.29. That’s not going to be enough to cover my shoe bill, but it should buy me some beers, at least. So, for now, I’ll keep collecting and see how well the prediction bears out.