Survey of Probability Concepts: Definitions and Applications

1. Chapter 5 A Survey of Probability Concepts

2. Goals • Probability Concepts • Define probability • Understand the terms: • Experiment • Outcome • Event • Describe these approaches to probability: • Classical • Empirical • Subjective

3. Goals • Probability Concepts • Calculate probabilities applying these rules: • Rules of addition • Rules of multiplication • Define the terms: • Conditional probability • Joint probability • Contingency Tables • Use a tree diagram to organize and compute probabilities • Principles of Counting

4. Types Of Statistics • Inferential Statistics • A decision, estimate, prediction, or generalization about a population, based on a sample • (Second part of definition of statistics) • Also known as: • Statistical inference • Inductive statistics

5. Statistical Inference OrInferential Statistics • Computing the chance that something will occur in the future! • This means that we will have to make decisions with incomplete information • Seldom does a decision maker have complete information from which to make a decision: • Marketers taking samples about a product name • Tests for wire tensile strength • Which player should the Mariners draft? • Should the soap opera Days of Our Lives be discontinued immediately? • Should I marry Jean?

6. Future Uncertainty • “Because there is uncertainty in decision making, it is important that all the known risks involved be evaluated scientifically” • Probability Theory will help: • Decision makers with limited information analyze the risks and minimize the inherent gamble

7. Define Probability • “Chance,” Likelihood,” Probability” • A number between zero & one, inclusive, describing the relative possibility (chance or likelihood) an event will occur in the future • Decimal or fraction: .25 = ¼, etc. • 0 ≤ P (x) ≤ 1 x means “the event” P means probability

8. Define Probability A value of zero means it cannot happen A value near zero means the event is not likely to happen A value of one means it is certain to happen A value near one means it is likely Probability Is the probability that a World Series will happen in 2006 close to one or to zero? Is the probability that a company will name a new breakfast cereal “Crud That Hurts Your Tummy” close to one or to zero?

9. Understand The Terms: • Experiment • Doing something and observing the one result • Outcome • A particular result of the experiment • Event • A collection of one or more outcomes of an experiment

10. Define: Experiment • A process that leads to the occurrence of one and only one of several possible observations • Example: roll die & there are 6 possible outcomes • Ask 250 Highline students whether they drink coffee • An experiment has two or more possible results (outcomes), and it is uncertain which will occur • An experiment is the observation of some activity or the act of taking some measurement

11. Define: Outcome • An outcome is the particular result of an experiment • Examples: • When you toss a coin, the possible outcomes are: • Heads • Tails • When you survey 1000 people and ask whether they will vote for candidate 1 or candidate 2, some of the possible outcomes are: • 455 would vote for candidate 1 • 592 would vote for candidate 1 • 780 would vote for candidate 1…

12. Define: Event • When one or more of the experiment’s outcomes are observed, we call this an event! • An event is the collection of one or more outcomes of an experiment • Example: • Roll die: • An even number can be an event • Boomerang tournament: • More than ½ the participants earned more than 60 points in the Trick Catch event • Political poll: • Less than 50% of those polled said they would vote for candidate A

13. Experiment, Outcome, Event

14. Definitions • Sample Space • A representation (list) of all possible outcomes in an experiment • It can be hard to list all the outcomes

15. Venn Diagram Sample Space Event A P(A) =.2  A • Sample Space is all outcomes = P(x) = 1 • Compliment P( A) = 1 – P(A) = 1 – .2 = .8 • P(A) + P(~A) = 1 or P(A) = 1 - P(~A)

16. Ch 5 Mutually Exclusive: The occurrence of one event means that none of the others can occur at the same time Two events, A & B, are mutually exclusive if both events, A & B, cannot occur at the same time In Venn Diagrams, there is no intersection: Examples: Die tossing experiment: the event “an even number” and the event “an odd number are mutually exclusive If you get an odd, it cannot also be even You can not have a product come off the assembly line that is both defective and satisfactory Define: Mutually Exclusive Even Odd

17. Definitions • Collectively Exhaustive • At least one of the events must occur when an experiment is conducted • If an experiment has a set of events that include every possible outcome, such as the events “an even number” and “an odd number,” then the set of events is collectively exhaustive • Mutually Exclusive & Collectively Exhaustive • If a set of events is mutually exclusive & collectively exhaustive, then the sum of the probabilities are equal to 1

18. Definitions • Independent • Events are independent if the occurrence of one event does not affect the occurrence of another (sample space is not changed) • The roll of a six, does not affect the next roll • P(B│A) = P(B) • Dependent • Events are dependent if the occurrence of one event affects the occurrence of another event (sample space is changed) • The chances of pulling a heart from a deck of cards? 13/52. But if you don’t put the card back (without replacement), what is the probability that you pull a heart next time? It depends: • 13/51 or 12/51

19. Definitions • Conditional Probability • The probability of a particular event occurring, given that another event has occurred • The sample space will change • The probability of the event B given that the event A has occurred is written P(B|A) • In the heart example, 13/51 or 12/51 are conditional probabilities Line means “given that.” “Probability that B will occur given that A has already occurred”

20. Sample Space Has a DVD player Has a TV Definitions • Joint Probability • A joint probability measures the likelihood that two or more events will happen concurrently • An example would be the event that a student has both a DVD Player and TV in his or her dorm room Root probabilities times conditional probabilities equal joint probabilities (Tree Diagrams)

22. Classical Approach To Probability The Classicaldefinition applies when there are n equally likely outcomes • Each outcome must have the same chance of occurring (fairness) • Events must be mutually exclusive & collectively exhaustive

23. Classical Approach To Probability A fair die is rolled once. • The experiment is rolling the die. • The possible outcomes are the numbers 1, 2, 3, 4, 5, and 6. • An event is the occurrence of an even number. That is, we collect the outcomes 2, 4, and 6.

24. Classical Approach To Probability • We do not need to conduct experiments to determine the probability under the classical approach? No! • Cards, dice, taxes • Example: • If three million returns are sent to your district office and 3000 will be audited the probability that you will be audited is:

25. Empirical Approach To Probability • The empirical definition applies when the number of times the event happens (in past) is divided by the number of observations • The probability of an event happening is the fraction of the time similar events happened in the past. • “Relative Frequency” • Law of Large Numbers: Over a large number of trials the empirical probability of an event will approach its true probability. This law allows us to use relative frequencies to make predictions.

26. Empirical Approach To Probability Throughout her teaching career Professor Jones has awarded 186 A’s out of 1,200 students. What is the probability that a student in her section this semester will receive an A? • To find the probability a selected student will earn an A: Based on past experience, we can estimate that the probability that a student will receive an A grade in a future class is .155

27. Subjective Approach To Probability • Subjective probability • There is little or no past experience on which to base probability • An individual assigns (estimates) a probability based on whatever information is available • Examples: • Estimate the probability that the Mariners will the World Series next year • Estimate the probability that AOL will merge with GOOGLE • Estimate the probability that a particular corporation will default on a loan • Estimate the probability mortgage rates will top 8 percent

28. Probability • P(x) is never known with certainty • P(x) is an estimate of an event that will occur in the future • There is great latitude in the degree of uncertainty surrounding this estimate • The degree of uncertainty is primarily based on the knowledge possessed by the individual concerning the underlying process • We know a great deal about rolling die • The underlying process is straight forward • We may not know much about whether a merger between companies will occur • Only some parts of the underlying process are known

29. Probability • The same laws of probability will be used regardless of the level of uncertainty surrounding the underlying process • Individuals will assign probabilities to events of interest • The difference amongst them will be in their confidence in the precision of the estimate

30. Sample Space Event A Has a DVD player Event B Has a TV player In this circumstance, Events A & B are not mutually exclusive!

31. Calculate Probabilities Applying These Rules: • Rules of addition • Rules of multiplication

32. Event A P(A) Event B P(B) Rules Of Addition Why? Don’t want to count twice! P(X)  1 HW #22, page 152 “at least one” = “either or”

33. Rules Of Addition Mutually Exclusive! Event B P(B) Event A P(A)

34. Rules Of Addition Example 1 • New England Commuter Airways recently supplied the following information on their commuter flights from Boston to New York:

35. Rules Of Addition Example 1

36. Rules Of Addition Example 2 • In a sample of 500 students: • 320 said they had a music sound system P(S) • 175 said they had a TV P(TV) • 100 said they had both P(S and TV) • 5 said they had neither TV 175 Both 100 S 320 In this circumstance, Events P(S and TV)are not mutually exclusive! What is the sample space?

37. Rules Of Addition Example 2 • If a student is selected at random, what is the probability that the student has: • Only a music sound system • Only a TV • Both a music sound system and TV P(S) = 320/500 = .64 P(TV) = 175/500 = .35 P(S and TV) = 100/500 = .20

38. Rules Of Addition Example 2 • If a student is selected at random, what is the probability that the: • Student has only a music sound system or TV? • Student has both a music sound system and TV? P(S or TV) = P(S) + P(TV) - P(S and TV) = 320/500 + 175/500 – 100/500 = .79 P(S and TV) = 100/500 = .20

39. Special Rule Of Multiplication • The special rule of multiplication requires that two events A and B are independent • Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other

40. Special Rule Of MultiplicationExample 1 • Chris owns two stocks: • IBM • General Electric (GE) • The probability that IBM stock will increase in value next year is .5 • The probability that GE stock will increase in value next year is .7 • Assume the two stocks are independent • What is the probability that both stocks will increase in value next year? • P(IBM and GE) = (.5)(.7) = .35

41. Special Rule Of Multiplication Example 2 • If the probability of selecting a finished boomerang with a blemish in the paint job is .02, what is the probability of randomly selecting four boomerangs from the production line (boomerangs just rolling off the line) and finding all four blemished? • Because there are so many, we can assume independence • P(selecting 1 boom with blemish) = .02 • P(selecting 4 booms with blemish) = .02*.02*.02*.02=.000000160

42. General Multiplication Rule • The general rule of multiplication is used to find the joint probability that two events will occur • It states that for two events A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of B given that A has occurred

43. General Multiplication Rule Example 1 • Now you have ten boomerangs and two of them have blemishes • We want to select one after the other • What is the probability of selecting a blemished boom followed by another blemished boom? • The sample space will change (without replacement) • The second P(X) is dependent on the first • P(Bblemish1)* P(Bblemish2) = 2/10*1/9 = 2/90  .0222 • Example 2: • In class example with women & men: what is the probability of selecting from a hat the names of three women • 10/20*9/19*8/18 = .105263158

44. In Class

45. A contingency table is used to classify observations according to two identifiable characteristics. Contingency tables are used when one or both variables are nominally or ordinally scaled. A contingency table is a cross tabulation that simultaneously summarizes two variables of interest.

46. General Multiplication Rule Example 2Contingency Table (Cross-classified)

47. General Multiplication Rule Example 2Contingency Table (Cross-classified) 80/200 35/200 Addition Rule: P(Would Not Remain or Has Less Than 1 Year Experience) = 80/200 + 35/200 – 25/200 = 90/200 = .45 P(Select 1-5 Years Experience) = 45/200 P(Would Not Remain given that 1-5 Years) = 15/45

48. Use A Tree Diagram To Organize And Compute Probabilities • Each segment of the tree is one stage in the problem • The branches of a tree diagram are weighted by probabilities Steps: • Draw heavy dots on left to represent the root of the tree • Two main branches are drawn with “root probabilities” • Create branches for each conditional probability • Write out Joint Probabilities

49. Draw Heavy Dots On Left To Represent The Root Of The Tree & Draw Two Main Branches With “Root Probabilities”

50. Create Branches For Each Conditional Probability