Lecture 8Lecture 8——Probability and Statistics (Ch. 3)Probability and Statistics (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 All of chapter 3 (pages 52 --64)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, 20: 8, 10, 16, 18, 20Homework assignments available on web pageHomework assignments available on web pageExam 1: two weeks from today, Fri. Feb. 8th (in class)Exam 1: two weeks from today, Fri. Feb. 8th (in class)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.Let W = number of possible outcomes (ways)Assign probability pito the ithoutcome11&1iiippWWW==×=∑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 iiththoutcome occurs outcome occurs nniitimestimeslimiiNnpN→∞⎛⎞=⎜⎟⎝⎠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:012345-3.0-2.5-2.0-1.5-1.0-0.50.0Nσ10.510 0.15100 0.041000 0.013210000 0.00356100000 0.00145log(σ )log(N)()()log log0.516aNbaσ=+=−Statistical fluctuationsStatistical fluctuations1/2Nσ−∝The axioms of probability theoryThe axioms of probability theory1. pi≥ 0, i.e. piis positive or zero2. pi≤ 1, i.e. piis less than or equal to 13. For mutually exclusive events, the probabilities for compound events, i and j, add()ijijppp+=+••Compound events, (Compound events, (ii+ + jj): this means either event ): this means either event iioccurs, or event occurs, or event jjoccurs, or both.occurs, or both.••Mutually exclusive: events Mutually exclusive: events iiand and jjare said to be mutually exclusive are said to be mutually exclusive if it is impossible for both outcomes (events) to occur in a sinif it is impossible for both outcomes (events) to occur in a single gle trial.trial.The axioms of probability theoryThe axioms of probability theory1. pi≥ 0, i.e. piis positive or zero2. pi≤ 1, i.e. piis 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 rrmutually exclusive events, the probability that one mutually exclusive events, the probability that one of the of the rrevents occurs is given by:events occurs is given by:12........rppp 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:11 16,666 36p=×=•Truly independent events always satisfy this property.•In general, probability of occurrence of r independent events is:12........rppp p=××