Probability J.R. Norris March 9, 2017 1
Contents 1 Mathematical models for randomness 6 1.1 A general definition 6 1.2 Equally likely outcomes. 6 1.3 Throwing a die 7 1.4 Balls from a bag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 A well-shuffled pack of cards . . . . 7 1.6 Distribution of the largest digit 7 1.7 Coincident birthdays 8 2 Counting the elements of a finite set 9 2.1 Multiplication rule 9 2.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 10 2.3 Subsets . 2.4 Increasing and non-decreasing functions . 10 2.5 Ordered partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Stirling's formula 11 3.1 Statement of Stirling's formula . 11 3.2 Asymptotics for log(n!) 11 3.3 Proof of Stirling's formula (non-examinable) 11 4 Basic properties of probability measures 13 4.1 Countable subadditivity 13 4.2 Continuity . 13 4.3 Inclusion-exclusion formula . 14 4.4 Bonferroni inequalities 15 5 More counting using inclusion-exclusion 16 5.1 Surjections . 16 5.2 Derangements 16 6 Independence 18 6.1 Definition . . . . . . . . . . Pairwise independence does not imply independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.2 18 6.3 Independence and product spaces 18 7 Some natural discrete probability distibutions 20 7.1 Bernoulli distribution 20 7.2 Binomial distribution 20 7.3 Multinomial distribution 20 7.4 Geometric distribution 21 7.5 Poisson distribution . 21 2
8 Conditional probability 22 8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8.2 Law of total probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 . 8.3 Bayes' formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8.4 False positives for a rare condition . 23 8.5 Knowledge changes probabilities in surprising ways 23 8.6 Simpson's paradox 24 9 Random variables 25 9.1 Definitions 25 9.2 Doing without measure theory 26 9.3 Number of heads . . . . . . . . . . . . . . . . . . . . .