Aftermath
The Math of Life
For Aftermath readers, here are some complementary resources that might be useful.
If you’re a footnote hound, we’ve put all footnotes here, online, organized by chapter. As you read the hard copy, you can have this section open on a nearby device and quickly pivot to a footnote, explore it online, and not lose track of your spot in the book. Seems like the sensible way to handle the manuscript’s many footnotes.
Confidence Interval Challenge: Chapter 4, Probability, includes a most revealing Confidence Interval challenge. Here’s the QR code that takes you to an online interactive version — a very simple and helpful way to take this on. But if you prefer, you can jot down your ranges on your hard copy of the book, then go to this section to track your ranges against the actuals. Tally up how many of your ranges spanned the actual value, and you’ll gain insight into whether you tend to over- or under-estimate what you know about uncertain phenomena.
Causality Challenge: Chapter 5, Prediction, ends by pointing you to some phenomena that have correlated data, challenging you to think through what lies behind the correlation — coincidence, true causality, or a shared underlying cause. Go here to find links to the relevant study or article for each phenomenom.
Systems Models: Chapter 5, Prediction, takes us to middle-school kids in Tuscon, Arizona, who have created a systems dynamics model for predicting future population levels. For those inclined, here’s the math behind such models. Not for everyone, but . . .
Life Impact Curve: Chapter 8, Decisions, introduces the concept of a Life Impact Curve. Here is an approach you can use to sketch out your own Life Impact Curve, and calibrate your risk aversion. It’s unlikely you’ll use this curve in making your decisions, but the process can be eye-opening.
For those who want help on formatting math problems for resources like Wolfram-Alpha, here’s our An Idiot’s Guide to Using a Math Smartphone App
Life Impact Curve
Population Dynamics Systems Model
Confidence Intervals
Causation versus Correlation Challenge
Life Impact Curve
Start with this blank Life Impact graph:
[here, we have a graph with the vertical axis being “Life Impact” and the horizontal axis being “Financial Shifts”]
Let’s translate shifts in your financial status into some qualitative measure of the Life Impact (LI) of these shifts. We start by arbitrarily assigning the 1.0 to a +$10,000 Financial Shift.
Now, the most you would risk for a 50/50 chance of winning $10,000 is _____ . [J.T., we map this to -1.0 on the LI graph]. Keep in mind that you’d certainly jump at this proposition if the downside risk were $500. And it’s unlikely you’d risk more than $5,000 for a 50/50 chance to win $10,000.
Next, the most you’d risk for a 90% chance of winning $10,000 is ______ .” Note that this response helps us sketch out your LI Curve using this equation:
90%*(1.0) = 10%*(-9.0)
Ping-ponging ahead, what gain would barely entice you to make a 50/50 bet with a loss that corresponds to the financial shift you came up with in Step 3 for your -9 LI value?
Calibrate your curve by considering how much you’d pay to insure against a .1% chance of losing $1,000,000.
Let’s say you’d pay $2,000 to insure against a .1% chance of a million-dollar loss. Say your QL curve maps a $2,000 decrease in your financial status to a -.3 QL value. Then we know that a loss of $1,000,000 maps to -300 on your LI Curve. This trade-off is invaluable in analyzing insurance policies that protect you from low-risk high-consequence catastrophes.
Population Dynamics Systems Model
A very simple model starts by assuming there are three population segments:
C = Number of Age 0-20 Children
P = Number of Age 21-40 Adults
E = Number of Age 41+ Adults
We make some crude assumptions. With each passing year, some 5% of the current year’s C leave the cohort – mostly the 20-year-olds who age into the P group. The P population produces babies, and we’ll assume that the number of new additions to the C cohort is about 15% of the current P population. We estimate that 6% of the P group leave the cohort each year – mostly 40-year-olds aging into the E group, with some dying. The P group get replenished by the 5% of the C’s who turn 21. Finally, about 2.5% of the E’s pass away each year. The dynamic equations governing this simple system are:
C (N +1) = .95*C(N) + 1.15*P(N)
P (N + 1) = .94*P(N) + .05*C(N)
E (N + 1) = .975*E(N) + .05*P(N)
Confidence Intervals
Phenomena Actual
# of American casualties in the Revolutionary War 6,800
# of U.S. labor strikes during World War II 14,000
# of stars in the Milky Way 100 billion
# of lakes (2+ acres of water surface) in Canada 880,000
# of ants on earth 20 quadrillion
# of U.S. schoolkids eligible for free or reduced lunch in 2019 29.6 million
# of U.S. Children Age 3-10 experiencing homelessness in 2017 600,000
Monthly wage (in U.S. $’s) of garment workers in Ethiopia $26/month
# of Instagram followers of Selena Gomez 396 million
Average # of standardized tests taken by a U.S. kid during their K12 years 112
# of Americans relying on food banks in 2021 53 million
# of Abortions performed in the U.S. in 2020 930,000
% of Princeton students from families in the bottom 20% of Income 2.2%
# of U.S. adults who believe the earth is flat 25.8 million
The # of active websites globally 200 million
# of U.S. citizens age 62 and older w/ outstanding student loan debt 1.4 million
# of millionaires in the U.S. in 2020 22 million
Money made in 2022 by nine-year-old Ryan Kajifrom his kids’ toys YouTube channel $29.5 million
Cost of the USS Gerald Ford aircraft carrier $13 billion
# of assault weapons in the U.S. 20 million
# of civilian-owned firearms in the U.S. 393 million
Idiot’s Guide
Here's a clear, beginner-friendly Math Entry Cheat Sheet that shows how to enter common math expressions into online tools like Wolfram|Alpha or MATLAB. It covers key concepts, syntax examples, and tips to avoid confusion—perfect for anyone just getting started.
Ballpark Estimates
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