Probability Chapter55 Random Experiments Probability Rules of Probability Independent Events Contingency Tables Counting Rules Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

A random experiment is an observational process whose results cannot be known in advance.A random experiment is an observational process whose results cannot be known in advance. The set of all outcomes (S) is the sample space for the experiment.The set of all outcomes (S) is the sample space for the experiment. A sample space with a countable number of outcomes is discrete.A sample space with a countable number of outcomes is discrete. Sample Space Sample Space Random Experiments 5-2

An event is any subset of outcomes in the sample space.An event is any subset of outcomes in the sample space. A simple event or elementary event, is a single outcome.A simple event or elementary event, is a single outcome. A discrete sample space S consists of all the simple events (E i ):A discrete sample space S consists of all the simple events (E i ): S = {E 1, E 2, …, E n } Events Events Random Experiments 5-3

The probability of an event is a number that measures the relative likelihood that the event will occur.The probability of an event is a number that measures the relative likelihood that the event will occur. The probability of event A [denoted P(A)], must lie within the interval from 0 to 1:The probability of event A [denoted P(A)], must lie within the interval from 0 to 1: 0 < P(A) < 1 If P(A) = 0, then the event cannot occur. If P(A) = 1, then the event is certain to occur. Definitions Definitions Probability 5-4

Three approaches to probability: ApproachExample EmpiricalThere is a 2 percent chance of twins in a randomly- chosen birth. What is Probability? What is Probability? ProbabilityClassicalThere is a 50 % probability of heads on a coin flip. SubjectiveThere is a 75 % chance that England will adopt the Euro currency by

The complement of an event A is denoted by A′ and consists of everything in the sample space S except event A.The complement of an event A is denoted by A′ and consists of everything in the sample space S except event A. Complement of an Event Complement of an Event Rules of Probability 5-6

unionThe union of two events consists of all outcomes in the sample space S that are contained either in event A or in event B or both (denoted A  B or “A or B”).  may be read as “or” since one or the other or both events may occur. Union of Two Events Union of Two Events (Figure 5.5) Rules of Probability 5-7

The intersection of two events A and B (denoted A  B or “A and B”) is the event consisting of all outcomes in the sample space S that are contained in both event A and event B.The intersection of two events A and B (denoted A  B or “A and B”) is the event consisting of all outcomes in the sample space S that are contained in both event A and event B.  may be read as “and” since both events occur. This is a joint probability. Intersection of Two Events Intersection of Two Events Rules of Probability 5-8

The general law of addition states that the probability of the union of two events A and B is:The general law of addition states that the probability of the union of two events A and B is: P(A  B) = P(A) + P(B) – P(A  B) When you add the P(A) and P(B) together, you count the P(A and B) twice. So, you have to subtract P(A  B) to avoid over-stating the probability. A B A and B General Law of Addition General Law of Addition Rules of Probability 5-9

Events A and B are mutually exclusive (or disjoint) if their intersection is the null set (  ) that contains no elements.Events A and B are mutually exclusive (or disjoint) if their intersection is the null set (  ) that contains no elements. If A  B = , then P(A  B) = 0 In the case of mutually exclusive events, the addition law reduces to:In the case of mutually exclusive events, the addition law reduces to: P(A  B) = P(A) + P(B) Mutually Exclusive Events Mutually Exclusive Events Rules of Probability Special Law of Addition Special Law of Addition 5-10

The probability of event A given that event B has occurred.The probability of event A given that event B has occurred. Denoted P(A | B). The vertical line “ | ” is read as “given.”Denoted P(A | B). The vertical line “ | ” is read as “given.” for P(B) > 0 and undefined otherwise Conditional Probability Conditional Probability Rules of Probability 5-11

The odds in favor of event A occurring isThe odds in favor of event A occurring is The odds against event A occurring isThe odds against event A occurring is Odds of an Event Odds of an Event Rules of Probability 5-12

Event A is independent of event B if the conditional probability P(A | B) is the same as the marginal probability P(A).Event A is independent of event B if the conditional probability P(A | B) is the same as the marginal probability P(A). Independent and Dependent Events When P(A) ≠ P(A | B), then events A and B are dependent.When P(A) ≠ P(A | B), then events A and B are dependent. Multiplication Law for Independent Events Multiplication Law for Independent Events 5-13

A contingency table is a cross-tabulation of frequencies into rows and columns. Example below.A contingency table is a cross-tabulation of frequencies into rows and columns. Example below. From the table, one can compute marginal probabilities, conditional probabilities, and check for independence between the two variables.From the table, one can compute marginal probabilities, conditional probabilities, and check for independence between the two variables. Contingency Table What is a Contingency Table? What is a Contingency Table? 5-14

If event A can occur in n 1 ways and event B can occur in n 2 ways, then events A and B can occur in n 1 x n 2 ways.If event A can occur in n 1 ways and event B can occur in n 2 ways, then events A and B can occur in n 1 x n 2 ways. In general, m events can occur n 1 x n 2 x … x n m ways.In general, m events can occur n 1 x n 2 x … x n m ways. Counting Rules Fundamental Rule of Counting Fundamental Rule of Counting 5-15

A permutation is an arrangement in a particular order of randomly sampled items from a group (i.e. XYZ is different from ZYX).A permutation is an arrangement in a particular order of randomly sampled items from a group (i.e. XYZ is different from ZYX). Counting Rules Permutations Permutations Combinations Combinations A combination is an arrangement of items chosen at random where the order of the selected items is not important (i.e., XYZ is the same as ZYX).A combination is an arrangement of items chosen at random where the order of the selected items is not important (i.e., XYZ is the same as ZYX). 5-16