AMS 102.7 Spring 2006 Jingyu Zou Elements of Statistics Lecture Notes # 12 1 Sample space and events De¯nition 1.1 Sample Space is the set of all possible outcomes of an experiment, denoted by ­. Example 1.2 If the experiment is simply tossing a coin, then ­ = fH; T g where H denotes head and T denotes tail. Example 1.3 If we toss a die, then ­ = f1; 2; 3; 4; 5; 6g. Each number denotes a possible point from a single toss. Example 1.4 Toss two coins. ­ = fHH;HT;TH;TT g. The outcome will be HH if both coins come up heads; it will be HT if the ¯rst coin comes up head and the second comes up tial; it will be TH if the ¯rst comes up tail and the second comes up head; and it will be TT if both comes up tails. De¯nition 1.5 Any subset E of the sample space ­ is known as an event. Example 1.6 In example 1.2, if E = fHg, then E is the event that a head appears on the flip of the coin. Similarly, if E = fT g, then E would be the event that a tail appears. Example 1.7 In example 1.3, if E = f1g, then E is the event that one appears on the toss of the die. If E = f2; 4; 6g, then E would be the event that an even number appears on the toss. Example 1.8 In example 1.4, if E = fHH; HT g, then E is the event that a head appears on the ¯rst coin. For any two events E and F of a sample space ­, E [ F is the event consisting of all points which are either in E or in F or in both E and F . That is, the event E [ F will occur if either E or F occurs. For any two events E and F , E \ F is the event consisting of all points which are both in E and F . That is, the event E \ F will occur only if both E and F occurs. E \ F can be simply written as EF . If EF = ;, then E and F are said to be mutually exclusive. For any event E, EC is the event consisting of all points in the sample space ­ which are not in E. That is EC will occur if and only if E does not occur. 1 2 Probability Consider a sample space ­. For each event E of the sample space ­, we assume that a number P (E) is de¯ned as satis¯es the following three conditions: (1) 0 · P (E) · 1 for any event E (2) P (­) = 1 (3) For any sequence of events E1;E2;:::;En which are mutually exclusive, that is, events for which Ei \ Ej = ; when i 6= j, then P (E1 [ E2 [ ::: [ En) = P (E1) + P (E2) + ::: + P (En) Corollary 2.1 (1) The probability of empty event is 0. That is, P (;) = 0 (2) If event E is a subset of event F , that is, E ½ F , then P (E) · P (F ) 2.