Level 8: Continuous chance and Bayes

Probability · level 8 of 8 · rough age 17 and beyond (a guide for parents; anyone can start anywhere)

The last level reaches first-year university probability. It starts with Bayes' theorem, the rule for learning from evidence, and the reason most positive results for a rare disease are false alarms. Then chance becomes continuous: probability as area under a curve, the normal distribution and its use for counts, and the two great theorems that explain why averages settle down and why bell curves are everywhere. It ends with random processes that unfold step by step, and with using random numbers to compute things that are not random at all.

Lesson	You will be able to
Bayes' theorem	Reverse a conditional probability with natural frequencies, a tree and the formula, and explain the base-rate fallacy
Continuous random variables	Find probabilities as areas under a density, explain why \(P(X = 3) = 0\), and use the uniform and exponential distributions
The normal distribution	Use z-scores, tables or a calculator to find normal probabilities and work backwards from a probability to a value
The normal approximation to the binomial	Approximate binomial probabilities with a normal curve and the continuity correction, and say when it works
The law of large numbers and the central limit theorem	Prove Chebyshev's inequality and the law of large numbers, and use the CLT for sums and averages
Random walks and Markov chains	Analyse a random walk, solve the gambler's ruin by first-step analysis, and find the long-run behaviour of a Markov chain
Monte Carlo methods	Estimate \(\pi\), areas and integrals with random numbers, judge their accuracy, and prove Buffon's needle result \(\tfrac{2}{\pi}\)

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