In a thought experiment proposed by the Italian probabilist Bruno de Finetti in order to justify Bayesian probability, an array of wagers is coherent precisely if it does not expose the wagerer to certain loss if his opponent is prudent. In philosophy, physics, and other fields, a thought experiment (from the German Gedankenexperiment) is an attempt to solve a problem using the power of human imagination. ... Bruno de Finetti (Innsbruck, June 13, 1906 - Rome, July 20, 1985) was an Italian probabilist and statistician, noted for the operational subjective conception of probability. ... In the philosophy of mathematics Bayesianism is the tenet that the mathematical theory of probability is applicable to the degree to which a person believes a proposition. ...

Operational subjective probabilities as wagering odds

You must set the price of a promise to pay $1 if John Smith wins tomorrow's election, and $0 otherwise. You know that your opponent will be able to choose either to buy such a promise from you at the price you have set, or require you to buy such a promise from him/her, still at the same price. In other words: you set the odds, but your opponent decides which side of the bet will be yours. The price you set is the "operational subjective probability" that you assign to the proposition on which you are betting. The set of prices is therefore coherent when they satisfy the probability axioms and related results such as the inclusion-exclusion principle. The probability P of some event E, denoted , is defined with respect to a universe, or sample space , of all possible elementary events in such a way that P must satisfy the Kolmogorov axioms. ... In combinatorial mathematics, the inclusion-exclusion principle states that if A1, ..., An are finite sets, then where |A| denotes the cardinality of the set A. For example, taking n = 2, we get a special case of double counting: in words, we can count the size of the union of sets...

"Dutch books"

A very trivial Dutch book

The rules do not forbid you to set a price higher than $1, but if you do, your prudent opponent may sell you that high-priced ticket, and then your opponent comes out ahead regardless of the outcome of the event on which you bet. Neither are you forbidden to set a negative price, but then your opponent may make you pay him to accept a promise from you to pay him later if a certain contingency eventuates. Either way, you lose. These lose-lose situations parallel the fact that a probability can neither exceed 1 nor be less than 0. In gambling a Dutch book or lock is a set of odds and bets which guarantees a profit, no matter what the outcome of the gamble. ...

A somewhat less trivial and more instructive Dutch book

Now suppose you set the price of a promise to pay $1 if the Boston Red Sox win next year's World Series, and also the price of a promise to pay $1 if the New York Yankees win, and finally the price of a promise to pay $1 if either the Red Sox or the Yankees win. You may set the prices in such a way that

But if you set the price of the third ticket too low, your prudent opponent will buy that ticket and sell you the other two tickets. By considering the three possible outcomes (Red Sox, Yankees, some other team), you will see that regardless of which of the three outcomes eventuates, you lose. An analogous fate awaits you if you set the price of the third ticket too high relative to the other two prices. This parallels the fact that probabilities of mutually exclusive events are additive (see probability axioms). In logic, two mutually exclusive (or mutual exclusive according to some sources) propositions are propositions that logically cannot both be true. ... The probability P of some event E, denoted , is defined with respect to a universe, or sample space , of all possible elementary events in such a way that P must satisfy the Kolmogorov axioms. ...

A person who has set prices on an array of wagers in such a way that he or she will suffer a net loss regardless of which outcome eventuates is said to have made a Dutch book. In gambling a Dutch book or lock is a set of odds and bets which guarantees a profit, no matter what the outcome of the gamble. ...

Conditional wagers and conditional probabilities

Now imagine a more complicated scenario. You must set the prices of three promises:

• to pay $1 if the Red Sox win tomorrow's game; the purchaser of this promise loses his bet if the Red Sox do not win regardless of whether their failure is due to their loss of a completed game or cancellation of the game, and

• to pay $1 if the Red Sox win, and to refund the price of the promise if the game is cancelled, and

• to pay $1 if the game is completed, regardless of who wins.

Three outcomes are possible: The game is cancelled; the game is played and the Red Sox lose; the game is played and the Red Sox win. You may set the prices in such a way that

(where the second price above is that of the bet that includes the refund in case of cancellation). Your prudent opponent writes three linear inequalities in three variables. The variables are the amounts he will invest in each of the three promises; the value of one of these is negative if he will make you buy that promise and positive if he will buy it from you. Each inequality corresponds to one of the three possible outcomes. Each inequality states that your opponent's net gain is more than zero. A solution exists if and only if the determinant of the matrix is not zero. That determinant is: In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

Thus your prudent opponent can make you a sure loser unless you set your prices in a way that parallels the simplest conventional characterization of conditional probability. This article defines some terms which characterize probability distributions of two or more variables. ...