Fall 2011 Homework 2: Question 1
(a) Arby has a belief system assigning a number $P_{Arby}(A)$ between 0 and 1 to every event A (for some sample space). This represents Arby's subjective degree of belief about how likely A is to occur. For any event A, Arby is willing to pay a price of $1000\cdot P_{Arby}(A)$ dollars to buy a certificate such as the one shown below:
Certificate
The owner of this certificate can redeem it for \$1000 if A occurs.
No value if A does not occur, except as required by federal,
state, or local law. No expiration date.

Likewise, Arby is willing to sell such a certificate at the same price. Indeed, Arby is willing to buy or sell any number of certificates at this price, as Arby considers it the "fair" price. Arby, not having taken Stat 110, stubbornly refuses to accept the axioms of probability. In particular, suppose that there are two disjoint events A and B with $P_{Arby}(A\cup B)\neq P_{Arby}(A)+P_{Arby}(B)$.
Solution: By buying/selling a sufficiently large number of certificates from/to Arby as described above, you can guarantee that you’ll get all of Arby’s money; this is called an arbitrage opportunity. This problem illustrates the fact that the axioms of probability are not arbitrary, but rather are essential for coherent thought (at least the first axiom, and the second with finite unions rather than countably infinite unions). Arbitrary axioms allow arbitrage attacks; principled properties and perspectives on probability potentially prevent perdition. Consult iTunes course for full solutions.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."