Today Today: Course Outline, Start Chapter 1 Assignment 1: –Read Chapter 1 by next Tuesday

Statistics What is Statistics? –Discipline that deals with the collection, summary, organization and interpretation of data –Used to help predict and answer questions about real processes

Example Dr. B. Spock was convicted of conspiracy during Vietnam era Convicted by an all-male jury Felt trial was unfair because no women were on the jury Jury list for Spock’s judge had 14.4% women Jury list for 6 other judges had 29% women Does this seem fair?

Example Questions: –What is the chance or probability that in a population with 29% women, we could select a jury pool of only 14.4% women? –Is 14.4% close to 29%? –What does chance or probability mean?

Sample Space The term experiment is used to denote an activity which produces an unpredictable outcome Will describe and summarize experiments using models Model describing an experiment should to take into account all possible outcomes Collection of all possible outcomes of an experiment is called the sample space and is denoted by

Example A balanced coin is flipped Do not know outcome in advance What if we tossed the coin 2 times? Example Roll of a fair die Example Number of people arriving in an emergency room in before one dies

Event A collection of outcomes (a subset of the sample space) is called an event An event, E, is said to occur if one of its outcomes occurs

Event Example A balanced coin is flipped two times Let E be the event that a tails occurs E = { } Example Committee of two people is chosen from 5 people (a-e) Let E be the event that person a is on the committee E = { }

Event Algebra We will generally introduce these ideas as needed

Event Algebra Example: Fair Die – E 1 E 3 ={ }

Venn Diagrams Can represent sample space and events using a Venn Diagram

Probability for Experiments A probability model assigns probabilities to outcomes and/or events in the sample space Simplest case is when the number of possible outcomes in the sample space is finite and each outcome is equally likely If there are N outcomes in the sample space and each outcome is equally likely then the probability of any individual outcome is 1/N Example: Fair die is rolled. What is the probability of observing 6?

Probability for Experiments An event E is a collection of outcomes from the sample space Recall, an event, E, is said to occur if one of its outcomes occurs So, P(E)=

Probability for Experiments Example: –In roulette, the possible outcomes are {00,0,1,2,…36} –Outcomes 00 and 0 are green, odd outcomes are red and even outcomes are black –Let R be the event that a red is observed in one spin of the roullette wheel –P(R)= Example: –In the committees example, the probability that person a is on the committee is…

Properties of Probability These are called the axioms of probability

Consequences of the Axioms

Example: –Small university offers 2 language courses, French and German –Everyone is required to take a language course –If 75% of students take French and 55% take German what proportion of students take both courses? –What proportion take only one course?

Combinatorics In the equally likely case, computing probabilities involves counting This can be hard…really Combinatorics is a branch of mathematics whichs develops efficient counting methods

Combinatorics Consider the rhyme As I was going to St. Ives I met a man with seven wives Every wife had seven sacks Every sack had seven cats Every cat had seven kits Kits, cats, sacks and wives How many were going to St. Ives? Answer:

Combinatorics