Mathematics for Better Decisions

Some mathematics makes you cleverer. Other mathematics keeps you from being fooled. This page orders the practical kind by its power to improve decisions, not by topic coverage, so that a learner meets compounding, probability and risk early and meets the specialised machinery later. The aim is not to become a human calculator but to see the mathematical structure beneath a choice.

Why order by decision-making power

Provisional (as a teaching judgment). A conventional syllabus follows the logic of the subject. A decision curriculum follows the cost of ignorance: which gaps most often lead people to mistake luck for skill, a small percentage for a small effect, or a rare disaster for an impossible one? The tiers below are an opinionated answer, and a learner may reorder them. All numbers on this page are generic examples and are not advice about any person's finances or health.

Tier 1: numerical survival

Fractions, percentages, ratios, percentage change, orders of magnitude, unit conversion and estimation. The goal is to notice quickly when a claim is unreasonable. Algebra then lets you turn a word problem into an equation and see how changing one variable moves another; see Algebra: The Skills That Carry Everything.

Tier 2: compounding

Demonstrated If a quantity grows by r per period, after n periods it is multiplied by (1 + r)^n. Doubling time is about ln 2 / ln(1 + r), and the Rule of 72 approximates it as 72 divided by the percentage rate: at 6 percent, about 12 periods (the exact value is 11.9). Exponential growth looks slow and then overwhelming, while linear growth adds the same amount each period. The same mathematics governs debt, populations, epidemics and adoption curves, and it fails when a limit is reached, so state the units and limits as in Quantitative Reasoning with Stated Assumptions.

Tier 3: probability

Independence, conditional probability, complements, base rates, expected value, and counting. The subject has a famous origin: in 1654 Pascal and Fermat corresponded about how to divide a stake in an unfinished game (the "problem of points"), a story told in Devlin's book.

Base rates in action. Suppose a condition affects 1 in 100 people; a screening test detects it 90 percent of the time and gives a false positive 9 percent of the time. Out of 10,000 people, 100 have the condition and 90 test positive; of the 9,900 without it, 891 also test positive. So only 90 out of 981 positives, about 9 percent, are true. Gigerenzer shows that stating problems as natural frequencies makes this much easier to grasp than stating probabilities.

Expected value is probability times payoff, summed. A fair die's expected value is 3.5. A bet can have positive expected value and still be unwise if one outcome is ruinous.

Permutations and combinations count outcomes: n! orderings of n items and C(n, k) subsets. These appear again in Discrete Mathematics and Graphs: Counting, Relations and Networks.

Tier 4: risk

Established Averages hide spread. Variance and standard deviation measure it. Distributions with fat tails produce extreme events far more often than a normal curve predicts (Taleb's popular treatment; the mathematics of heavy-tailed distributions is standard). Regression to the mean, first described by Galton (1886), means extreme observations tend to be followed by less extreme ones even when nothing has changed, and it is a frequent source of fake explanations for "improvement after a bad streak".

Risk of ruin. Losses are asymmetric: a 50 percent loss needs a 100 percent gain to recover. A repeated activity with a chance of permanent loss will eventually hit it, so survival comes before optimisation. The Kelly criterion (Kelly, 1956) gives a growth-optimal bet size under strong assumptions; treat it as a conceptual guide and not a recipe, since its inputs are rarely known.

Tier 5: statistical literacy

• Correlation versus causation: ask about confounders and about the direction of cause.
• Survivorship bias: in the standard telling of a Second World War story, the statistician Abraham Wald reasoned that armour should go where returning aircraft showed little damage, because planes hit there did not return. Mangel and Samaniego (1984) discuss Wald's actual survivability work, which was more technical than the popular anecdote, so treat the anecdote as an illustration.
• Relative versus absolute risk: doubling a one-in-a-million risk is still tiny. This is central in Reading Nutrition Evidence: What a Study Can and Cannot Show.
• False positives and negatives, sample size, and confidence intervals.

Bayesian updating, step by step. (1) State a prior probability. (2) Observe evidence. (3) Ask how much more likely the evidence is if each explanation is true. (4) Update in proportion. (5) Avoid jumping to certainty. Worked example: prior 10 percent, evidence three times likelier under the hypothesis. Prior odds 1:9 become 3:9, a posterior of 25 percent. This is the quantitative form of the reasoning in Judgment Under Uncertainty: Probability, Causation and Forecasting.

Tier 6: systems

Optimisation means maximising an objective subject to constraints, with marginal benefit, marginal cost and diminishing returns. Identify bottlenecks and single points of failure. Feedback loops with delays produce overshoot and oscillation; Meadows' Thinking in Systems is the accessible introduction, and Systems Thinking: Feedback, Queues, Information and Dynamics goes further. Trend over time, not a single noisy reading, is the right unit for judging any slow process, as discussed in Gradual Fat Loss and Maintenance: A Method.

Weekly practice

One suggested rhythm is three short sessions per concept:

1. Learn: study one idea and write a plain-language explanation.
2. Calculate: solve five to ten problems before reaching for a calculator.
3. Apply: use the idea on a real decision in your own field.

For each concept keep a card with: a definition, one equation, one worked example, one common mistake, and one application.

Ten core mental models

Compounding; expected value; base rates; risk of ruin; variance; opportunity cost; marginal analysis; exponential growth; Bayesian updating; feedback loops. Fact versus analogy: each is a precise mathematical idea; using one as a metaphor for a messy situation is analogy and should be labelled as such.

Try this

1. Take a claim from the news that quotes a percentage and convert it to natural frequencies (out of 1,000 people).
2. Compute how long a quantity growing at 4 percent takes to double with the Rule of 72, then check with logarithms.
3. Make a card for "base rate" and write one worked example with different numbers.

Further reading

• Gerd Gigerenzer, Calculated Risks (Simon & Schuster, 2002).
• Tversky and Kahneman, "Judgment under uncertainty: Heuristics and biases," Science 185, 1974.
• Donella Meadows, Thinking in Systems (Chelsea Green, 2008).
• Keith Devlin, The Unfinished Game (Basic Books, 2008).
• Mangel and Samaniego, "Abraham Wald's work on aircraft survivability," JASA 79, 1984.