Probability 101 CS 2800: Discrete Structures, Spring 2015 Sid Chaudhuri Thought for the Day #0 What are the chances Cornell will declare a snow day !ew "ork $oological Society Thought for the Day #1 %fter how many years will a monkey produce the Complete Works of Sha#espeare with more than 50% probability Thought for the Day #1 %fter how many years will a monkey produce the Complete Works of Sha#espeare with more than 50% probability )or just an intelligible tweet + ,lements o& Probability -heory ● .utcome ● Sa'ple Space ● ,/ent ● Probability Space 0iramax 2eads -ails playingintheworldgame3wordpress3co' .utco'es 2eads -ails playingintheworldgame3wordpress3co' Sample Space 2eads -ails playingintheworldgame3wordpress3co' Sample Space Set of all possible outcomes of an e1periment Some Sample Spaces ● Coin toss: { , } ● Die roll: { , , , , , } ● Weather: { , , , } Sample Space Set of all mutually exclusive possible outcomes of an e1periment Event Subset of sample space Some E/ents ● ,/ent of a coin landing heads: { } Some E/ents ● ,/ent of a coin landing heads: { } ● ,/ent of an odd die roll: { , , } Some E/ents ● ,/ent of a coin landing heads: { } ● ,/ent of an odd die roll: { , , } ● ,/ent of weather li#e 6thaca: { , , , } Some E/ents ● ,/ent of a coin landing heads: { } ● ,/ent of an odd die roll: { , , } ● ,/ent of weather li#e 6thaca: { , , , } ● ,/ent of weather li#e Cali&ornia: { } Careful! ● -he sa'ple space is a set (of outcomes) ● %n outcome is an element of a sa'ple space ● %n e/ent is a set (a subset of the sa'ple space) – 6t can be empty (the null e/ent { } or ∅, which ne/er happens+ – 6t can contain a single outcome (simple/elementary e/ent+ – 6t can be the entire sample space (certain to happen) ● Strictly speaking, an outcome is not an e/ent (it's not e/en an elementary e/ent+ Probability Space Sample space S … plus function P assigning real-/alued probabilities P(E) to e/ents E ⊆ S … satis&ying olmogorov!s axioms All three are needed! ;olmogorov's Axioms 1. <or any e/ent E, we ha/e P(E) ≥ 0 ;olmogorov's Axioms 1. <or any e/ent E, we ha/e P(E) ≥ 0 2. P(S) = 1 ;olmogorov's Axioms 1. <or any e/ent E, we ha/e P(E) ≥ 0 A countable set of events can be indexed by the natural numbers as “first event”, “second event”, “third event” and 2. P(S) = 1 so on. We'll see a formal definition of countability later, but for now you don't need to worry about this too much. =3 6f a countable set of e/ents E1, E2, E3, … are pairwise disjoint (>'utually exclusi/e”), then P(E1 ∪ E2 ∪ E3 ∪ …) = P(E1) + P(E2) + P(E3) + … ;olmogorov's Axioms 1. <or any e/ent E, we ha/e P(E) ≥ 0 A countable set of events can be indexed by the natural numbers as “first event”, “second event”, “third event” and 2. P(S) = 1 so on. We'll see a formal definition of countability later, but for now you don't need to worry about this too much. =3 6f a countable set of e/ents E1, E2, E3, … are pairwise disjoint (>'utually exclusi/e”), then P(E1 ∪ E2 ∪ E3 ∪ …) = P(E1) + P(E2) + P(E3) + … Can you pro/e these from the axioms 13 <or any e/ent E, we ha/e P(E) ≥ 0 ● In a valid probability 23 P(S) = 1 space (S, P) =3 6f e/ents E1, E2, E3, … are pairwise – P(E') = 1 – P(E) for any dis*oint (>'utually e1clusi/e?), then event E P(E1 ∪ E2 ∪ E3 ∪ …) = P(E1) + P(E2) + P(E3) + … Space for boardwor# Can you pro/e these from the axioms 13 <or any e/ent E, we ha/e P(E) ≥ 0 ● 6n a /alid probability 23 P(S) = 1 space (S, P) =3 6f e/ents E1, E2, E3, … are pairwise – P(E') = 1 – P(E) for any dis*oint (>'utually e1clusi/e?), then event E P(E1 ∪ E2 ∪ E3 ∪ …) = P(E ) + P(E ) + P(E ) + … – P(∅) = 0 1 2 3 Space for boardwor# ,@uiprobable Probability Space ● %ll outcomes equally li#ely (&air coin, fair die...) ● Aaplace's definition of probability (only in finite equiprobable spaceC+ |E| P(E)= |S| ,@uiprobable Probability Space ● %ll outcomes equally li#ely (&air coin, fair die...) ● Aaplace's definition of probability (only in finite equiprobable spaceC+ Number of elements |E| P(E)= (outcomes) |S| in E Number of elements (outcomes) in S P(event that sum is N) N -im Stellmach, Wikipedia Derolamo Cardano )1501915EF+ Aiar, gambler, lecher, heretic saderpoGglogster.