Basic Probability ----------------- Def: A (finite) *probability space* is a finite set \Omega together with a function \mu: \Omega -> [0,1] having the property that \sum \mu(x) = 1 x \in \Omega We call \Omega the "sample space" and we call \mu the "probability measure". Eg 1: You roll a fair die six times: \Omega = {1,2,3,4,5,6} \mu(1)=\mu(2)=...=\mu(6)=1/6 Eg 2: You deal out five cards from a deck of cards: \Omega = { 5-element subsets of the 52 cards } \mu(x) = 1 / C(52,5) for all x \in }omega Eg 3: You flip an unfair coin 10 times. The coin lands heads a fraction p=0.51 of the time: \Omega = {0,1}^10 \mu(x) = p^{#1(x)} (1-p)^{#0(x)} where #1(x) = the number of 1-bits in the string x and #0(x) = the number of 0-bits in the string x. Def: Let (\Omega, \mu) be a probability space. An *event* A is a subset of \Omega. Def: Let A be an event of probability space (\Omega, \mu). Pr[A] = \sum \mu(x) x \in A "The probability of event A" Eg: Returning to Eg 1, "you roll an even number" is an event -- the event is A = {2,4,6}. Pr[A] = 3 * (1/6) = 1/2. In general, whenever you hear "probability" make sure that you are clear WHAT is the probability space and WHAT is the event in question. Propositions: - Pr[\emptyset] = 0 // by definition - Pr[\Omega] = 1 - Pr[A] + Pr[\Omega -A] = 1 - If A and B are disjoint events (that is, disjoint sets) then Pr[A u B] = Pr[A] + Pr[B] - ("sum bound") Pr[A u B] <= Pr[A] + Pr[B] - In general, Pr[A u B] = Pr[A] + Pr[B] - Pr[A intersect B] // inclusion-exclusion principle