YMS Chapter 6 Probability: Foundations for Inference 6.1 – The Idea of Probability

Probability – 3 Interpretations ► Any outcome of any random phenomenon is the proportion of times it would occur in a very long series of repetitions ► Long-term relative frequency ► Branch of math that describes the pattern of chance outcomes

► When individual outcomes are uncertain but there is still a regular distribution of outcomes in the long run ► Relative frequencies of outcomes seem to settle down to fixed values in the long run ► Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run Randomness – 3 Interpretations

Exploring Randomness ► Must have a long series of independent trials ► Probability is empirical (based on previous experience) ► Computer simulations are very useful 6.1 Practice – p334 #6.4, 6.9 and 6.10

YMS 6.2 Probability Models

► Sample space (S)  The set of all possible outcomes ► Event  Any outcome or set of outcomes of a random phenomenon  A subset of S ► Probability model  A mathematical description of a random phenomenon consisting of two parts: S and the assignment of probabilities to events ► Multiplication (Counting) Principle  Multiply number of outcomes for each event to find total number of ways  Use a tree diagram to visually represent and find sample space

► Any probability is a number between 0 and 1.  0 < P(A) < 1 ► All possible outcomes together must have probability 1.  P(S) = 1 ► Complement Rule - The probability that an event does not occur is 1 minus the probability that it does.  P(A C ) = 1- P(A) Probability Rules

► Disjoint or Mutually Exclusive Events  If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.  P(A or B) = P(A) + P(B) – P(A and B) ► Independent events  Knowing that if one event occurs it does not change the probability that the other occurs  P(A and B) = P(A)P(B)

Basic Set Theory ► Union  Combination of elements in sets ► Intersection  What the sets have in common ► Null set  Set without elements ► Venn Diagrams  Very useful to create when answering questions about relationships among sets 6.2 Practice – p340 #6.15, 6.19, 6.26, 6.28, 6.29, 6.35, 6.42

YMS 6.3 General Probability Rules

► Rule for disjoint events  P(one or more of A, B, C) = P(A) + P(B) + P(C) ► Two events are independent if P(B|A)=P(B) ► Conditional Probability  The probability of one event under the condition that we know another event

► General Multiplication Rule (rewrite for conditional probability)  P(A and B) = P(A)*P(B|A) ► Bayes’s Rule  Don’t memorize!  Use Tree Diagrams 6.3 Practice – p365 #6.51, 6.52, 6.58, 6.59, 6.61, 6.64, 6.65