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Book 12 · Probability and statistical inference · Rung 1Book 12
For Doctors

Obesity Expertise · Book 12

Probability and statistical inferenceRung 1 · Introductory

Probability, random variables, and the difference between a parameter and an estimate

Dr. Harsh Maheshwari · Version 1.0 · updated 13 October 2026 · CC BY-NC-SA 4.0

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Why this book

A positive test result usually feels like an answer. Often it is not, and most doctors asked to work out what it means give a number far too high. This book teaches the rules of probability that settle the question, ending at Bayes’ theorem, and the handful of patterns of chance that keep turning up in health data. Then it separates three things a reported number can be: a fact about everyone, an estimate from a sample, or a prediction. Every later book on statistics leans on these ideas, and so does every conversation with a patient about a test.

How to read it

Read Book 0 and Book 2 first, or at least the sections this book names as it goes. Then read this one in order, with a pen and a calculator, and do the practice problems in each section before you read on. The book asks you to build one thing: a one-page handout explaining a positive result for a test used in your own clinic. Choose the test and find your clinic’s register before you start, because one section asks you to count how common the condition is among your own patients.

Before you start

Book 0, in particular percentages (A4), ratios and proportions (A5), powers and roots (A6), and all of Part D: what a probability is and odds (D1), independence (D2), the two-way table with sensitivity, specificity and the positive predictive value (D3), mean and standard deviation (D5) and parameter against statistic (D6). Book 2, sections 1, 4 and 13 to 15: the symbols and the Σ sign (1), e and natural logarithms (4), and the random variable, expectation and variance (13 to 15). The mathematics is arithmetic with a calculator, fractions, powers, the sum sign, and one-line algebra; no calculus is needed beyond reading an area under a curve as a probability, which Book 2 taught.

1 Sample space, events and the three rules every probability obeys

2 Conditional probability as a definition, and the multiplication rule

3 Independence given a condition, and where it fails

4 The law of total probability: a probability as a weighted average over groups

5 Bayes’ theorem, from the multiplication rule, on a diagnostic test

6 The base-rate case that defeats most clinicians

7 Natural frequencies: Bayes by counting people

8 The odds form of Bayes’ theorem and the likelihood ratio

9 The prosecutor’s fallacy: P(evidence | innocent) is not P(innocent | evidence)

10 Where the prior comes from: pre-test probability in your own clinic

11 Counting arrangements: factorials and “n choose k”

12 The Bernoulli and binomial distributions: counting successes in independent trials

13 The geometric and hypergeometric distributions: waiting for the first success, and drawing without replacement

14 The Poisson distribution: counts of events at a steady rate

15 The normal distribution: what generates it, and areas under it

16 Skewed quantities and the log scale: when effects multiply

17 Covariance and correlation of two random variables

18 Sums and averages of many independent draws, simulated by hand

19 Parameter, statistic, estimator, estimate

20 How good is an estimator: bias and spread, found by listing every sample

21 The likelihood: the same formula read as a function of the parameter

22 Parameter, estimate or prediction: what a reported number is a number of

23 Worked journey: One clinic test, from the question to a handout a patient can read

References 207
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