Consistent personal betting rates satisfy the basic rules of probability. Consistency in this sense is called coherence.
Personal probabilities and betting rates are all very well. But thus far they have no structure, no rules–in fact, not much meaning. Now we give one argument that betting rates ought to satisfy the basic rules for probability. We have already had three thought experiments in Chapter 13. Here are two more.
FOURTH THOUGHT EXPERIMENT: SETS OF BETTING RATES
A group of beliefs can be represented by a set of betting rates.
Imagine yourself advertising a set of betting rates. For each of the propositions A, B, … K in the set, you offer betting rates pa, pb, pc , …, pk .
In this imaginary game, you are prepared to bet, say,
on A at rate pa , or
against A at rate (1–pa ).
FIFTH THOUGHT EXPERIMENT: SIMPLE INCONSISTENCY
These are personal betting rates. Couldn't you choose any fractions you like?
Of course. But you might be inconsistent.
For example, suppose you are concerned with just two possibilities, B (for “below zero”) and ~B:
B: On the night of next March 21, the temperature will fall below 0°C at the Toronto International Airport meteorological station.
~B: On the night of next March 21, the temperature will not fall below 0°C at the Toronto International Airport meteorological station.
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