# Chance and Uncertainty: Probability Theory

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1 Chance and Uncertainty: Probability Theory Formally, we begin with a set of elementary events, precisely one of which will eventually occur. Each elementary event has associated with it a probability, which represents the likelihood that a particular event will occur; the probabilities are all nonnegative, and sum to 1. For example, if a five-card hand is dealt from a thoroughly-shuffled deck of 52 cards, there are 2,598,960 different elementary events (different hands that might be dealt), each of which has probability 1/2,598,960 of occurring. An event is a collection of elementary events, and its probability is the sum of all of the probabilities of the elementary events in it. For example, there are 778,320 distinct five-card hands that contain precisely one ace. Therefore, the event A = the hand contains precisely one ace has probability 778,320/2,598,960 of occurring; this is written as Pr(A) = [In many applications, the set of elementary events is not made explicit. For example, we can consider Pr(the Cubs win the World Series) without needing to detail every possible way they might do so.] The shorthand A B is written to mean at least one of A or B occurs (more concisely, A or B ; more verbosely, the disjunction of A and B ), and A B is written to mean both A and B occur (more concisely, A and B ; more verbosely, the conjunction of A and B ). A sometimes-useful relationship is Pr(A) + Pr(B) = Pr(A B) + Pr(A B). Two events A and B are mutually exclusive (or disjoint) if it is impossible for both to occur, i.e., if A B =. The complement of A, written A c, is the event A does not occur. Obviously, Pr(A)+ Pr(A c ) = 1. Whenever Pr(A) > 0, the conditional probability that event B occurs, given that A occurs (or has occurred, or will occur), is Pr(B A) = Pr(A B)/Pr(A). Two events A and B are independent if Pr(A B) = Pr(A) Pr(B). Note that when A and B are independent, Pr(B A) = Pr(B). Three events, A, B, and C, are mutually independent if each pair is independent, and furthermore Pr(A B C) = Pr(A) Pr(B) Pr(C). Similarly, any number of events are mutually independent if the probability of every conjunction of events is simply the product of the event probabilities. [The following example illustrates why we must be careful in our definition of mutual independence: Let our elementary events be the outcomes of two successive flips of a fair coin. Let A = first flip is Heads, B = second flip is Heads, and C = exactly one flip is Heads. Then each pair of events is independent, but the three are not mutually independent.] 1

2 A collection of events A 1,..., A k is mutually exclusive and exhaustive if each is disjoint from the others, and together they cover all possibilities. If A 1,..., A k are mutually exclusive and exhaustive, then Pr(B) = Pr(B A 1 ) Pr(B A k ) = Pr(B A 1 ) Pr(A 1 ) Pr(B A k ) Pr(A k ). Often, one observes some consequence B of a chance event, and wishes to make inferences about the outcome of the original event A. In such cases, it is typically easier to compute Pr(B A), so the following well-known rule is of use. Bayes' Rule: Pr(A B) = Pr(A B)/Pr(B) = Pr(B A) Pr(A) / Pr(B). The result at the top of this page can be used to compute Pr(B). Example: On the basis of a patient's medical history and current symptoms, a physician feels that there is a 20% chance that the patient is suffering from disease X. There is a standard blood test which can be conducted: The test has a false-positive rate of 25% and a false-negative rate of 10%. What will the physician learn from the result of the blood test? Pr(X present test positive) = Pr(test positive X present) Pr(X present) / Pr(test positive) = / Pr(test positive), and Pr(test positive) = Pr(test positive X present) Pr(X present) + Pr(test positive X not present) Pr(X not present) = = 0.38, so Similarly, Pr(X present test positive) = 0.18/0.38 = Pr(X present test negative) = Pr(test negative X present) Pr(X present) / Pr(test negative) = / Pr(test negative) = 0.02/(1-0.38) =

3 The preceding calculations are much simpler to visualize in probability-tree form: Disease X present 0.2 test positive test negative = = 0.02 Disease X absent 0.8 test positive test negative = = 0.60 From the tree we can immediately write Pr(X present test positive) = 0.18 / ( ). 3

5 won proportionately split one share, and the bettors on the horse that finished second proportionately split the other share. Show bets that your horse will finish in the top three are handled similarly. Can you make money at the track? Of course you can, IF you are much better able to predict outcomes than most of the other bettors. (It s not enough to be just a bit better, since you must overcome the track s takeout.) Much statistical analysis has gone into the development of horserace betting systems. A good system must combine the ability to (probabilistically) predict race outcomes with the ability to determine when the posted odds differ enough from the prediction to offer a profit. A common feature of any sensible system is that it will often recommend not betting at all (when, for example, the posted odds are in line with the predictions of the system). One business application of parimutuel betting is to forecasting: If you wish to derive a consensus forecast of the likelihood of one or the other of several events occurring, you could endow each of your forecasters with a budget, and require that they all place bets on the outcome. The resulting odds could then be taken as their actual (combined) forecast. Goldman Sachs and Deutsche Bank have recently opened an online enterprise where bettors can wager on such economic derivatives as the inflation rate, housing starts, or new unemployment claims. The odds that result can be used as a composite (across the bettors) estimate of the economic future Game Odds A common marketing technique is to offer consumers participation in a collection game. McDonald s, for example, runs an annual Monopoly game in conjunction with Hasbro (the publisher of the board game). Stickers representing Monopoly properties are attached to food-product containers, and a customer who collects some particular combination of properties over the duration of the game can claim a prize (usually a free McDonald s or Hasbro product). McDonald s benefits directly from heightened consumer interest, and Hasbro indirectly by building name recognition for its product line. A legal requirement (in most states) of such games is that the odds of winning the various prizes must be posted. The standard language states, If you collect k stamps, the odds of winning prize P are 1 in x. This translates directly to the odds against winning are x-1:1. 5

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