Probability Theory Topics Basic Probability Concepts Sample Spaces

Probability Theory

Topics • Basic Probability Concepts: Sample Spaces and Events, Simple Probability, and Joint Probability, • Conditional Probability • Bayes’ Theorem • Probability Distribution

Sample Spaces Collection of all Possible Outcomes e. g. All 6 faces of a die: e. g. All 52 cards of a bridge deck:

Events • Simple Event: Outcome from a Sample Space with 1 Characteristic e. g. A Red Card from a deck of cards. • Joint Event: Involves 2 Outcomes Simultaneously e. g. An Ace which is also a Red Card from a deck of cards.

Visualizing Events • Contingency Tables Ace Black Red Total • 2 2 4 Not Ace 24 24 48 Total 26 26 52

Simple Events The Event of a Happy Face There are 5 happy faces in this collection of 18 objects

Joint Events The Event of a Happy Face AND Light Colored 3 Happy Faces which are light in color

Special Events Null event Club & diamond on 1 card draw Complement of event For event A, All events not In A: Null Event

Dependent or Independent Events The Event of a Happy Face GIVEN it is Light Colored E = Happy Face Light Color 3 Items: 3 Happy Faces Given they are Light Colored

Contingency Table Red Ace A Deck of 52 Cards Ace Not an Ace Total Red 2 24 26 Black 2 24 26 Total 4 48 52 Sample Space

Contingency Table 2500 Employees of Company ABC Agree Neutral Opposed | MALE 900 200 400 | FEMALE 300 100 600 | Total 1200 300 1000 | Sample Space Total 1500 1000 2500

Tree Diagram Event Possibilities Full Deck of Cards Red Cards Ace Not an Ace Black Cards Not an Ace

Probability • Probability is the numerical measure of the likelihood that the event will occur. • Value is between 0 and 1. • Sum of the probabilities of all mutually exclusive and collective exhaustive events is 1. 1 Certain . 5 0 Impossible

Computing Probability • The Probability of an Event, E: P(E) = = Number of Event Outcomes Total Number of Possible Outcomes in the Sample Space X T e. g. P( ) = 2/36 (There are 2 ways to get one 6 and the other 4) • Each of the Outcome in the Sample Space equally likely to occur.

Computing Joint Probability The Probability of a Joint Event, A and B: P(A and B) = Number of Event Outcomes from both A and B Total Number of Possible Outcomes in Sample Space e. g. P(Red Card and Ace) =

Joint Probability Using Contingency Table Event B 1 Event B 2 Total A 1 P(A 1 and B 1) P(A 1 and B 2) P(A 1) A 2 P(A 2 and B 1) P(A 2 and B 2) P(A 2) Total Joint Probability P(B 1) P(B 2) 1 Marginal (Simple) Probability

Computing Compound Probability The Probability of a Compound Event, A or B: e. g. P(Red Card or Ace)

Contingency Table 2500 Employees of Company ABC Agree Neutral Opposed | MALE 900 200 400 | FEMALE 300 100 600 | Total 1200 300 1000 | Sample Space Total 1500 1000 2500

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0. 24

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0. 24 2. Neutral

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0. 24 2. Neutral 300/2500 = 0. 12

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0. 24 2. Neutral 300/2500 = 0. 12 3. Opposed to the proposal, GIVEN that the employee selected is a female

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0. 24 2. Neutral 300/2500 = 0. 12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0. 60

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0. 24 2. Neutral 300/2500 = 0. 12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0. 60 4. Either a female or opposed to the proposal

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0. 24 2. Neutral 300/2500 = 0. 12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0. 60 4. Either a female or opposed to the proposal ………. . 1000/2500 + 1000/2500 - 600/2500 = 1400/2500 = 0. 56

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0. 24 2. Neutral 300/2500 = 0. 12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0. 60 4. Either a female or opposed to the proposal ………. . 1000/2500 + 1000/2500 - 600/2500 = 1400/2500 = 0. 56 5. Are Gender and Opinion (statistically) independent?

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0. 24 2. Neutral 300/2500 = 0. 12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0. 60 4. Either a female or opposed to the proposal ………. . 1000/2500 + 1000/2500 - 600/2500 = 1400/2500 = 0. 56 5. Are Gender and Opinion (statistically) independent? For Opinion and Gender to be independent, the joint probability of each pair of A events (GENDER) and B events (OPINION) should equal the product of the respective unconditional probabilities…. clearly this does not hold…. . check, e. g. , the prob. Of MALE and IN FAVOR against the prob. of MALE times the prob. of IN FAVOR …they are not equal…. 900/2500 does not equal 1500/2500 * 1200/2500

Compound Probability Addition Rule P(A 1 or B 1 ) = P(A 1) +P(B 1) - P(A 1 and B 1) Event B 1 B 2 Total A 1 P(A 1 and B 1) P(A 1 and B 2) P(A 1) A 2 P(A 2 and B 1) P(A 2 and B 2) P(A 2) Total P(B 1) P(B 2) 1 For Mutually Exclusive Events: P(A or B) = P(A) + P(B)

Computing Conditional Probability The Probability of Event A given that Event B has occurred: P(A B) = e. g. P(Red Card given that it is an Ace) =

Conditional Probability Using Contingency Table Conditional Event: Draw 1 Card. Note Kind & Color Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 Revised Sample Space

Conditional Probability and Statistical Independence Conditional Probability: P(A B) = Multiplication Rule: P(A and B) = P(A B) • P(B) = P(B A) • P(A)

Conditional Probability and Statistical Independence (continued) Events are Independent: P(A B) = P(A) Or, P(B A) = P(B) Or, P(A and B) = P(A) • P(B) Events A and B are Independent when the probability of one event, A is not affected by another event, B.

Bayes’ Theorem P(Bi A) = Same Event Adding up the parts of A in all the B’s

Bayes’ Theorem Given a hypothesis H and an event E P(H|E) = P(E|H) x P(H) P(E) Where P(E) = P(E|H) x P(H) + P(E| H) x P( H)

Example You have a physical complaint which is one of the symptoms of a rare and unpleasant disease. 1 in 500 people have the disease. The probability of having the physical symptom if you have the disease is 0. 7. The probability of having the symptom without also having the disease is 0. 01. How likely is it that you are suffering from the disease?

Probability Distribution Models Probability Distributions Continuous Discrete