Chapter 3 - Probability Theory Notes

CH3: Probability theory An event is defined to be any subset of the sample space of S. The empty set is the set containing no elements. and S are both subsets of S and thus events. ] [ is called an Impossible event and S is called a certain event An elementary event is an event with exactly one member. Kolmogorov's axioms of probability Suppose that S is the sample space for a random experiment. For all events ACS, we define the probability with the following properties of A denoted Pr (A), to be a real number O Pr (A) SI for all A CS Pr (s) 1 If AnBØ (le if A and B Pr (AUB) - Pr (A) + Pr (B) are mutually exclusive events) then → by consequence, Pr (p) - 0 Functions usually look like y=f(x) you put in number (x), you get another number (y) out. P(A) is a new kind of function you put a set (A) in, and out pops a number between zero and one Relative probability We start by doing n trials of the random experiment and counting the number of times r that some event ACS occurs during n trials. Then we define r/n to be the relative frequency of the event A. Obviously, or/n≤ 1. Thus relative frequencies and probabilities both lie between zero and one. We can think of the probability of the event A as the relative frequency of A as n, the number of trials of the random experiment gets very large Pr (A) 248 lim As the number of trials increases, the relative frequency tends to get closer and closer to the "true" probability. Some useful theorems theorem 1 proof Let ACS Then Pr (A)-1-Pr (A) we write S as the union of the mutually exclusive events AUA= because S A and A are mutually exclusive, le ANA we can use axiom 3 to State Pr(AUA) but AUA So Pr (A) + Pr (A) S, and Pr (s)=1, by Kalmogorov's 2nd axiom Pr (AUA)-1 Pr (A) + Pr (A)=1 Therefore, and Pr (A)1 Pr (A) theorem 2 If ACS and BCS then Pr(A) Pr (AUB) + Pr(AUB) proof write A as the union of the two clearly, mutually exclusive gets: A=(AnB) u (AnB). (ΑΛΒ) (ΑΠB) = Ø therefore using axiom notice that theorem 2 3, Pr (A)= Pr (AMB)+Pr (AnB) can also be expressed as Pr (AnB) Pr (A) - Pr (AnB) theorem 3 the addition rule → Pr (AUB) = arbitrary events A and B, Pr (A) + Pr (8) Pr (AnB) proof write AUB AUB as the union of two mutually exclusive sets. BU (ANB) because B and AnB apply axiom 3 and say are mutually exclusive, we can again Pr (AUB) Pr (B) + Pr (AUB). But, by theorem 2, Pr (AUB) Pr (A) Pr (AB) theorem 4 if BCA, then Pr (B) Pr (A). proof → If B C A then we can write mutually exclusive sets, A as the union of two A=BU (ANB) and Pr (A) Pr (B) + Pr (AMB) because Pr (AЛВ)20 > Pr (B) as all probabilities are non negative. theorem 5 If A,, Az,..., An are pairwise mutually exclusive le then Ain Aj = for i #j Pr (A. U Az U or in more concise notation, U An). Pr (Ai) + Pr (As) +...+ Pr (An) P. (Ai) Pr (Ai) Equally probable elementary events In theory, all probability problems can be solved using theorem 5. The elementary events whose union make up the sample space S are always mutually exclusive because if one elementary event occurs, no other elementary event occurs. Therefore, if we knew the probabilities of all the elementary events, we would also be able to compute the probabilities of any event in S. By theorem 5, the probability of any event is Simply the sum of the probabilities of the elementary events that make up the event. There is a wide class of problems for which we do know the probabilities of all the elementary events in a sample Space. These are the problems for which it is reasonable to assume that all the elementary events are equally likely. If there are N elementary events contained in S, and each one has the same probability of occuring, then the probability of each and every elementary event must be 1/N. Equally probable elementary events occur in many games of chance. When a card is drawn from a pack of 52 playing cards, the probability of any particular cord is assumed to be 1/52 Let A be some event in this scenario. Then A must consist of the union of elementary events, each of probability 1/N. If we could determine the number of elementary events contained in A, then we could write Pr (A ) number of elementary events contained in A number of elementary events contained in S n(A) (A) N n(s) Permulations and combinationg In problems in which all elementary events are equally probable, it is usually impractical to list and Count all elementary events contained in sample space or in event of interest. Theory of combinations and permutations comes to rescue and events can be determined quite easily. Theory is summarized Counting rules"