Concepts of ProbabilityTrial question: we are given a die. How can we determine theprobability that any given throw results in a six?Try doing many tosses:• Plot cumulative proportion of sixes• Also look at other features that may or may not indicatefairness, e.g. are there three consecutive sixes in 100 throws?1Law of Large NumbersOver very many trials, the proportion of times a given outcomeoccurs becomes very close to the true probability of that out-come.It’s a useful way to check up on whether probabilities are correct,but it can also serve as a definition of probability.Independent trials:Successive trials of some random event (e.g. tosses of a coin,throws of a die) are said to be “independent” if the outcome ofany one trial does not affect the outcomes of any others.2Defining Probabilities• Relative frequency definition based on the proportion oftimes an event occurs in a large population. Implicit use ofLaw of Large Numbers.• Equally likely definition applies when there are a finite num-ber of possible outcomes and there is no reason to assumeone occurs more frequenlty than others. Examples: throwinga die, drawing cards from a deck.• Subjective probability definition applies when we use ourown judgment to assess probabilities. Examples include bet-ting on the outcome of a sporting event or the kinds ofjudgments business executives make all the time.3Calculating ProbabilitiesIn this section we describe some of the methods for calculat-ing probabilities and illustrate some of the mathematical issuesassociated with finding probabilities of complicated events.Sample SpaceThe first step is to define the sample space. This is just a listof all the possible outcomes of an experiment.4Example 1. One die, possible outcomes {1, 2, 3, 4, 5, 6}.Example 2. Two dice. We could list the possible outcomes bygiving the total for each die, e.g. (3, 4) would mean that thefirst die showed a 3 while the second showed a 4. Then thesample space would be all 36 possible combinations — (1,1),(1,2), (1,3), and so on all thr way through to (6,6).Alternatively, we might just be interested in the total. In thiscase the sample space is {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.Note, however, that whereas in the first case we could still as-sume that all 36 outcomes are equally likely, in the second casethis would not be true. That might be an argument for preferringthe first way of writing the sample space.5Example 3. Sample space for a multi-choice quiz (see text formore discussion):{CCC,CCI,CIC,CII,ICC,ICI,IIC,III}.6Example 4. Suppose we were interested in all possible outcomesof the 2009 UNC football season (see table). How would wedefine the sample space in this case?Opponent H/ADuke HMiami HVirginia HFlorida State HThe Citadel HEast Carolina HGeorgia Tech AVirginia Tech ANC State ABoston College AConnecticut A7For example, we could define it just be number of wins, so thesample space is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.Or we could define the sample space by the win-loss record, e.g.W LW LLW LW LW L would be one of 2048 possible outcomes.Although we would not in practice try to write out the completesample space, there is no conceptual problem about defining thesample space in this way.Or, we could go further and try to list even more information,e.g. the actual scores in each of the games. This would lead toa very large (hypothetically infinite) sample space, but again, solong as we made it clear what possible outcomes are permitted,there is nothing conceptually difficult about defining the samplespace in this way.8EventsAn event is technically defined to be any subset of the samplespace. Usually events are denoted by capital letters, A, B, etc.Two possible events are• A: UNC wins at least five football games, and• B: In two throws of a die, the total is at least 10.Need to represent an event as a subset of the sample space.With UNC football games, if we define sample spaceS = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}then the event A isA = {5, 6, 7, 8, 9, 10, 11}9With dice, if we represent sample space asS = {(1, 1), (1, 2), (1, 3), ...., (6, 6)}.then the subset that defines the event B isB = {(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)}.10Defining ProbabilitiesIn the case of equally likely outcomes, the probability of the eventE is defined asP (E) =number of outcomes in event Enumber of outcomes in sample space S.For example, in the case of the event B defined above,P (B) =636=16.UNC football example: no reason to assume that all 12 possibleoutcomes are equally likely. However if we assigned them prob-abilities p0, p1, ..., p11(by whatever means) we can write casethatP (A) = p5+ p6+ p7+ p8+ p9+ p10+ p11+ p12.11The moral: We can build up probabilities for complicated eventsby starting with probabilities for elementary events and then ma-nipulating them according to the rules of probability. However,we still need to think carefully about those probabilities for ele-mentary