Probability, Bayesian reasoning, and uncertainty

Reason from outcome spaces to Bayesian updating, sampling uncertainty, statistical tests, and explicit decision losses.

Prerequisite: Arithmetic and percentages. No statistics or calculus required. Fourteen lessons introduce counting, conditioning, independence, Bayesian evidence, expectations, distributions, sampling, intervals, tests, multiple comparisons, and decisions. Seven interactive experiments and 28 worked exercises lead to a complete screening-policy audit.

Lessons

1. 1

   Define the experiment before assigning probabilities

   An outcome, an event, and a probability model answer different questions.

2. 2

   Count paths without double-counting

   Multiplication counts sequential choices; addition combines disjoint cases.

3. 3

   Conditioning changes the denominator

   Restrict the population before calculating the fraction.

4. 4

   Separate independence from mutual exclusion

   Independent events leave each other’s probabilities unchanged. Exclusive events cannot happen together.

5. 5

   Update a base rate with evidence

   Bayes’ rule compares the routes that could have produced the observed result.

6. 6

   Accumulate evidence without counting it twice

   Likelihood ratios multiply odds, subject to the right conditioning assumptions.

7. 7

   Calculate expectations without promising outcomes

   An expectation is a probability-weighted average, not necessarily a possible or typical result.

8. 8

   Describe spread, tails, and continuous values

   Equal means can hide different risks. Density height is not point probability.

9. 9

   Model repeated samples, not just individual outcomes

   The sampling distribution describes how an estimate changes across hypothetical repeated datasets.

10. 10

    Distinguish confidence from posterior probability

    An interval’s interpretation comes from the procedure and probability model that produced it.

11. 11

    Interpret tests, p-values, and power

    A test asks how the data compare with a specified null model. It does not directly assign a probability to that model.

12. 12

    Account for many chances to find a result

    Multiple comparisons, repeated peeking, and selective reporting change what a striking result means.

13. 13

    Choose actions using explicit losses

    The most probable state does not always determine the least harmful action.

14. 14

    Audit a screening-and-action proposal

    Connect base rates, evidence quality, sampling uncertainty, and consequences in one complete case.