Driving near sirens while Black

Jadarrius Rose and Fermi estimation

A few weeks ago, Jadarrius Rose, a 23-year-old truck driver, was attacked by a police dog from the Circleville, Ohio, Police Department during a highway chase initiated due to a missing mudflap on his trailer. Rose, who attempted to surrender with his hands up, was bitten by the dog despite orders from another officer not to release the dog. The horrifying ordeal, caught on police body-camera, showed Rose in a defenseless position, pleading with the dog to stop as it continued to bite into his left arm. Rose’s attorney, Ben Crump, said:

Imagine the psychological trauma and the mental anguish that Black people go through in America when sirens and flashing lights come on behind them. That’s what the Jadarrius Rose video represents to every Black person in America, the fear of being the next hashtag.

This got me thinking. How many Black people in America hear a siren while in a car? I wanted to try to answer this by doing a Fermi problem. Get ready, because for the first time in this Substack, I am going to give you a math lesson.

What is a Fermi problem?

Fermi problems require estimation and approximation to solve questions that often seem impossible due to the lack of immediate data. This method of approximation is attributed to the physicist Enrico Fermi, who was renowned for his knack for performing accurate estimations using minimal or no hard data. A common phrase you might hear to describe a Fermi problem is “back of the envelope estimate.” Arguably, the most famous Fermi problem is “How many piano tuners are there in Chicago?”

To solve a Fermi problem, you need to know a few key concepts: powers, logarithms, and orders of magnitude.

To start, powers involve multiplying a number, the base, by itself a certain number of times, determined by the exponent. For instance, ²³ equals 8 because 2 is multiplied by itself 3 times.

Logarithms are the inverses of powers. They provide the exponent you need to raise a given base to obtain a certain number. For instance, log₂8 equals 3, meaning you need to raise 2 to the power of 3 to get 8.

Understanding these concepts allows us to work with orders of magnitude. An order of magnitude is essentially the base-10 logarithm of a number, rounded to the nearest integer (positive or negative counting number or zero). It provides a way to approximate numbers by the nearest power of 10. For instance, the numbers 10 and 100 have orders of magnitude 1 and 2 respectively. What about 200? I’d say that the order of magnitude is also 2, since log₁₀200 rounds to 2. And then we can go backwards from the order of magnitude and find the “nearest” power of 10. For instance, the nearest power of 10 for 473 is 1000, as log₁₀473 rounds to 3, and 1⁰³ equals 1000.

If we are guessing orders of magnitude / powers of ten and we can’t choose between two successive ones, we settle for an order of magnitude that falls in-between. So, between orders of magnitude 1 and 2 (which represent 10 and 100), we’d use an order of magnitude of 1.5, which equates to roughly 30 (10 1.5≈ 31.6). This technique continues with numbers in higher ranges, for example, between 100 and 1000, the in-between value would be 300.

In summary, solving Fermi problems often involves rounding and approximating to powers of 10, or their in-between values, for simplicity. These are broad estimates. My experience teaching students Fermi problems is that the biggest challenge for them is just letting go of the details and embracing the fact that it is, by design, a rough estimate that they are making.

How many Black people hear a siren while in a car?

Here’s a very rough estimate.

Number of cars in America: Well, the U.S. has somewhere around 300 million people, and I think I’ve heard that there is approximately one car per person in America, so I will go with 300 million cars.

Percentage of cars on the road every day: One percent seems too low. Ten percent also seems to low to me. One hundred percent? Not sure. What’s in-between? That would be 30%. That seems low to me. I’m going to go with 100%, recognizing that the true figure is assuredly somewhat less.

Number of people per car: I’m going to just start guessing orders of magnitude. Is it one person per car? Maybe. Is it 10? No way. What about what’s in-between, namely, 3? That still seems too high to me. I’m going to assume one person per car, recognizing that the true figure is probably a bit more.

Percentage of drivers who are Black: An unforeseen advantage of choosing one person per car is that we don’t have to worry about the racial demographics of a group of passengers in each car. We can just guess what percentage of drivers are Black. Let’s make the (probably wrong, but simplifying) assumption that car ownership rates are the same across racial groups, so that the percentage of drivers who are Black will mirror the percentage of the U.S. that is Black. Then we can just ask, what percentage of the U.S. is black? It’s higher than 1%, higher than 10%, and lower than 100%. But it is also lower than what’s in-between, namely, 30%. So I’m going to go with 10%, which is a slight underestimate.

Probability of hearing a siren while driving: Here, I’m really grasping at straws. I’m just going to go through orders of magnitude. On a given day, do I think the probability of hearing a siren is 1%? It feels higher than that. Ten percent? Maybe. What about 100%? Definitely not. How about 30%? Maybe. If I am choosing between 10% and 30%, I feel like 30% is more accurate. If the probability of hearing a siren on a given day is 30%, that’s approximately equivalent to saying “you hear a siren while driving once every three days or so.”

Putting it all together: We have (300 million cars) x (100% of cars on the road) x (1 person / car) x (10% of people in cars who are Black) x (30% chance of hearing a siren per day) = 9 million Black people / day who hear a siren while in a car.

The quantitatively inclined among you may be balking. Maybe you disagree with my guesses. Maybe you are uncomfortable with the many, many simplifying assumptions. Remember: this is a Fermi problem. For any applied mathematicians, computational social scientists, engineers, or other researchers reading this, I think this problem would be really interesting to address in a more precise and accurate way with some agent-based modeling.

Impact of the trauma

Quantifying the negative impact of trauma, such as from hearing a police siren, is really hard, as it involves a range of factors that are both objective and subjective. Doing this is well outside of my realm of expertise, so I won’t try, but here are a few approaches that come to mind as possibilities.

Individual impacts

(1) Psychological research in which experts help evaluate the severity of a trauma response could lend some insight. There are standardized clinical scales like the Impact of Event Scale — Revised (IES-R) and the Posttraumatic Stress Disorder Checklist (PCL).

(2) Physiological measurements might have something to say. These would include measuring heart rate variability, cortisol levels (a stress hormone), or brain activity through EEGs or MRIs. A significant deviation from normal ranges could suggest a high level of stress or trauma.

(3) Quality of life assessments are validated questionnaires that can be used to assess an individual’s perceived quality of life. The WHO Quality of Life-BREF (WHOQOL-BREF) is a well-known tool that might have something to say.

(4) Economic impact. This can be quantified through lost productivity (missed work days, decreased performance, etc.), costs of therapy and medication, and costs of potential support needed (such as disability benefits).

Societal impacts

(1) If many people experience such trauma, the increased need for mental health services could put a strain on the healthcare system and lead to increased spending.

(2) If trauma leads to individuals being unable to work, or less productive at work, this can have economic implications for society.

(3) Trauma can hinder the cognitive development in children and adolescents, impacting their academic performance and long-term educational prospects.

In closing

We need to see Jadarrius Rose as an individual and we also need to recognize that his story gives a view into structural racism in this country. Quantitative thinking is one way of engaging with this view. Looking ahead, I’m wondering about policy interventions. Issues of over-policing aside (not that we can really put them aside), the exercise of writing this post has made me curious about whether we could simply get police to use sirens less. I’m sure people have thought and written about this, and I plan to look into the issue. If you have thoughts or pre-existing knowledge, please chime in in the chat.

Your neighbor,

Chad