Basic Probability — Deterministic versus Probabilistic

1 Basic Probability (pp. 377 391)70 Deterministic versus ProbabilisticDeterministic:All data is known beforehand Once you start the system, you know exactly what is going tohappen. the amount of money in a bank account. If you know the initial deposit, and the interest rate, then: You can determine the amount in the account after one :Element of chance is involved You know the likelihood that something will happen, but youdon t know when it will happen. a die until it comes up 5 . Know that in each roll, a 5 will come up with Probability 1/6. Don t know exactly when, but we can predict Probability (pp. 377 391)71 Basic ProbabilityDefinition:Anexperimentis any process whose outcome is :The set of all possible outcomes of an experiment iscalled thesample space, :Each outcomex Xhas a number between 0 and 1that measures its likelihood of occurring. This is theprobabilityofx, denotedp(x). a die is an experiment; the sample space is{}.

2 The individual probabilities are allp(i)=.Definition:AneventEis something that happens(in other words, a subset of the sample space).Definition:GivenE,theprobabilityo f the event (p(E)) is thesum of the probabilities of the outcomes making up the roll of the [is 5 ] or [is odd] or [is prime].. (E1)=,p(E2)=,p(E3)=. Basic Probability (pp. 377 391)72 Determining ProbabilitiesThree methods for determining the Probability of an occurrence: Relative frequency method:Repeat an experiment manytimes; assign as the Probability the fractionoccurrences# experiments a bulls-eye 17 times out of 100; set theprobability of hitting a bulls-eye to bep(bulls-eye) = Equal Probability method:Assume all outcomes haveequal Probability ; assign as the probability1# of possible side of a dodecahedral die is equally likely toappear; decide to setp(1) =112. Subjective guess method:If neither method above applies,give it your best likely is it that your friend will come to a party?

3 Basic Probability (pp. 377 391)73 Independent EventsDefinition:Two events areindependentif the probabilities ofoccurrence do not depend on one a red die and a blue die. Event 1: blue die rolls a 1. Event 2: red die rolls a events are independent. Event 1: blue die rolls a 1. Event 2: blue die rolls a events are a card, any card! Shuffle a deck of 52 cards. Event 1: Pick a first card. Event 2: Pick a second events wake up and don t know what day it is. Event 1: Today is a Event 2: Today is Event 3: Today is Modeling Probability (pp. 377 391)74 Independent Events When eventsE1(inX1)andE2(inX2)areindependente vents,p(E1andE2)=p(E1)p(E2). is the Probability that today is a cloudy weekday? When eventsE1(inX1)andE2(inX2)areindependente vents,p(E1orE2)=1 (1 P(E1))(1 P(E2))=P(E1)+P(E2) p(E1)p(E2)Proof:Venn diagram / is the Probability that you roll a blue 1 OR a red 6?This does not work Probability (pp. 377 391)75 Decision TreesDefinition:Amultistageexperiment is one in which each stage is asimpler experiment.

4 They can be represented using atree branch of the tree represents one outcomexof that level sexperiment, and is labeled byp(x). a biased 492/3HT 291/3T1/3TH 292/3TT 191/3 Independent or dependent? and SF State two soccer games. (p. 382)1: Ind122: Ind 383/42: SF 181/41: SF122: Ind 161/32: SF 132/3 Independent or dependent? Basic Probability (pp. 377 391)76 Expected value / mean Even with the randomness, what do you expect to happen? Suppose that each outcome in a sample space has a numberr(x)attached to it. (examples: number of pips on a die, amount ofmoney you win on a bet, inches of precipitation falling)This functionris called arandom :Theexpected valueormeanof a random variable is thesum of the numbers weighted by their probabilities. Mathematically, =E[X]=p(x1)r(x1)+p(x2)r(x2)+ +p(xn)r(xn).Idea:With probabilityp(x1), there is a contribution ofr(x1), many heads would you expect on average whenflipping a biased coin twice? many wins do you expect Indiana to have?

5 Basic Probability (pp. 377 391)77 Expected value / meanWhen two random variables are on twoindependentexperiments,the expected value operation behaves nicely:E[X+Y]=E[X]+E[Y]andE[XY]=E[X]E[Y] . throw a red die and a blue die. What is the expectedvalue of the sum of the dice and the product of the dice?b+r1234 5 61 2345 6 72 3456 7 83 4567 8 94 5678 9105 6789101167 8 9 10 11 12b r123456112345622468101233 6 9 12 15 184 4 8 1216202455 10 15 20 25 3066 12 18 24 30 36E[X+Y]=E[XY]=Component Reliability78 Component ReliabilityMany systems consist of components pieced together. To determinehow reliable thesystemis, determine how reliableeach componentis and apply Probability :Thereliabilityof a system is its Probability of the space shuttle into space with a 1 Stage 2 Stage 3 In order for the rocket to launch, LetR1= 90%,R2= 95%,R3= 96% be the reliabilities of Stages 1 (system success) =p(S1 success and S2 success and S3 success)Component Reliability79 Component with the space are two independent methods in which earth cancommunicate with the space shuttle A microwave radio with reliabilityR1= An FM radio, with reliabilityR2= In order to be able to communicate with the shuttle.

6 P(system success) =p(MW radio success or FM radio success)