Analysis Analysis is the examination of processes facts and data to gain an understanding of why problems occur and where opportunities for improvement exist Statistical inference is the process of drawing conclusions about unknown characteristics of a population from which data were taken Predictive statistics focus on cause and effect relationships and predictions of future performance from historical data 1 Basic Probability Concepts An experiment is a process that results in some outcome The outcome of an experiment is a result that we observe The collection of all possible outcomes of an experiment is called the sample space Probability is the likelihood that an outcome occurs 2 Probability Properties Label the n outcomes in a sample space as O1 O2 On where Oi represents the ith outcome in the sample space The probability associated with any outcome must be between 0 and 1 0 P Oi 1 for each outcome Oi The sum of the probabilities over all possible outcomes must be 1 0 P O1 P O2 P On 1 3 Events An event is a collection of one or more outcomes from a sample space If A is any event the complement of A denoted as Ac consists of all outcomes in the sample space not in A Two events are mutually exclusive if they have no outcomes in common 4 Calculating Probabilities Rule 1 The probability of any event is the sum of the probabilities of the outcomes that compose that event Rule 2 The probability of the complement of any event A is P Ac 1 P A 5 Calculating Probabilities Rule 3 If events A and B are mutually exclusive then P A or B P A P B Rule 4 If two events A and B are not mutually exclusive then P A or B P A P B P A and B 6 Conditional Probability Conditional probability is the probability of occurrence of one event A given that another event B is known to be true or have already occurred Multiplication rule of probability 7 Tree Diagram 8 Calculation of Joint Probabilities 9 Independent Events Two events A and B are independent if P A B P A 10