Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Lecture 8—Probability and Statistics Lecture 8—Probability and Statistics (Ch. 3)(Ch. 3)Friday January 25Friday January 25thth•Quiz on Chapter 2•Classical and statistical probability•The axioms of probability theory•Independent events•Counting eventsReading: Reading: All of chapter 3 (pages 52 - 64)All of chapter 3 (pages 52 - 64)Homework 2 due TODAYHomework 2 due TODAY***Homework 3 due Fri. Feb. 1st*******Homework 3 due Fri. Feb. 1st****Assigned problems, Assigned problems, Ch. 3Ch. 3: 8, 10, 16, 18, : 8, 10, 16, 18, 2020Homework assignments available on Homework assignments available on web pageweb pageExam 1: two weeks from today, Fri. Feb. 8th (in Exam 1: two weeks from today, Fri. Feb. 8th (in class)class)ClassicalClassicalThermodynamicsThermodynamicsClassical and statistical probabilityClassical and statistical probabilityClassical probability:•Consider all possible outcomes (simple events) of a process (e.g. a game).•Assign an equal probability to each outcome.Let W = number of possible outcomes (ways)Assign probability pi to the ith outcome1 1& 1i iip p WW W= = � =�Classical and statistical probabilityClassical and statistical probabilityClassical probability:•Consider all possible outcomes (simple events) of a process (e.g. a game).•Assign an equal probability to each outcome.Examples:Coin toss:Coin toss:WW = 2 = 2 ppii = 1/2 = 1/2Classical and statistical probabilityClassical and statistical probabilityClassical probability:•Consider all possible outcomes (simple events) of a process (e.g. a game).•Assign an equal probability to each outcome.Examples:Rolling a dice:Rolling a dice:WW = 6 = 6 ppii = 1/6 = 1/6Classical and statistical probabilityClassical and statistical probabilityClassical probability:•Consider all possible outcomes (simple events) of a process (e.g. a game).•Assign an equal probability to each outcome.Examples:Drawing a card:Drawing a card:WW = 52 = 52 ppii = 1/52 = 1/52Classical and statistical probabilityClassical and statistical probabilityClassical probability:•Consider all possible outcomes (simple events) of a process (e.g. a game).•Assign an equal probability to each outcome.Examples:FL lottery jackpot:FL lottery jackpot:WW = 20M = 20Mppii = 1/20M = 1/20MClassical and statistical probabilityClassical and statistical probabilityStatistical probability:•Probability determined by measurement (experiment).•Measure frequency of occurrence.•Not all outcomes necessarily have equal probability.•Make Make N N trialstrials•Suppose Suppose iithth outcome occurs outcome occurs nnii times timeslimiiNnpN��� �=� �� �Classical and statistical probabilityClassical and statistical probabilityStatistical probability:•Probability determined by measurement (experiment).•Measure frequency of occurrence.•Not all outcomes necessarily have equal probability.Example:lim 0.312iiNnpN��� �= �� �� �Classical and statistical probabilityClassical and statistical probabilityStatistical probability:•Probability determined by measurement (experiment).•Measure frequency of occurrence.•Not all outcomes necessarily have equal probability.More examples:Classical and statistical probabilityClassical and statistical probabilityStatistical probability:•Probability determined by measurement (experiment).•Measure frequency of occurrence.•Not all outcomes necessarily have equal probability.More examples:0 1 2 3 4 5-3.0-2.5-2.0-1.5-1.0-0.50.0N1 0.510 0.15100 0.041000 0.013210000 0.00356100000 0.00145log( )log(N)( ) ( )log log0.516a N bas = +=-Statistical fluctuationsStatistical fluctuations1/ 2Ns-�The axioms of probability theoryThe axioms of probability theory1. pi ≥ 0, i.e. pi is positive or zero2. pi ≤ 1, i.e. pi is less than or equal to 13. For mutually exclusive events, the probabilities for compound events, i and j, add( )i ji jp p p+= +•Compound events, (Compound events, (ii + + jj): this means either event ): this means either event ii occurs, or event occurs, or event jj occurs, or both. occurs, or both.•Mutually exclusive: events Mutually exclusive: events ii and and jj are said to be mutually exclusive are said to be mutually exclusive if it is impossible for both outcomes (events) to occur in a single if it is impossible for both outcomes (events) to occur in a single trial.trial.The axioms of probability theoryThe axioms of probability theory1. pi ≥ 0, i.e. pi is positive or zero2. pi ≤ 1, i.e. pi is less than or equal to 13. For mutually exclusive events, the probabilities for compound events, i and j, add•In general, for In general, for rr mutually exclusive events, the probability that one mutually exclusive events, the probability that one of the of the rr events occurs is given by: events occurs is given by:1 2........rp p p p= + + +Independent eventsIndependent eventsExample:What is the probability of What is the probability of rolling two sixes?rolling two sixes?Classical probabilities:Classical probabilities:166p =Two sixes:Two sixes:1 1 16,66 6 36p = � =•Truly independent events always satisfy this property.•In general, probability of occurrence of r independent events is:1 2........rp p p p= � �