Probability as a Measure of Uncertainty

Probability measures what is not known before an outcome is observed. Afterward, the outcome is no longer 65% YES: it is either YES or NO. The probability belonged to the information state before resolution, not to the resolved fact.

The 0-to-1 scale #

For an event $A$,

$$0\le P(A)\le1, \qquad P(A^c)=1-P(A).$$

$P(A)=0$ means impossible under the model, $P(A)=1$ means certain under the model, and values between them express uncertainty. $A^c$ is the complement: everything in the declared outcome space that is not $A$.

Two meanings often cooperate. A frequency interpretation says that among many comparable 65% forecasts, roughly 65% should occur. A degree-of-belief interpretation says 0.65 summarizes current evidence about one event. Neither makes the single outcome predictable with certainty.

More likely is not known #

If $P(A)=0.65$ and $P(A^c)=0.35$, then $A$ is more likely, but the less likely outcome still occurs often enough to matter. In 100 comparable independent cases, the expected counts are 65 YES and 35 NO:

$$E[N_{YES}]=100\times0.65=65.$$

The actual count need not equal 65. The expected count is a long-run center, not a schedule saying which 35 cases fail.

The running 65% trading example #

Let $q=0.65$ be a trader's probability that a binary contract pays 1. That estimate says nothing yet about whether the contract is worth buying. Value also depends on price. Buying at 0.50 and buying at 0.90 express the same belief but create radically different economics.

Application to no-hedge #

In a no-hedge strategy, the reference market provides evidence for a probability; it does not guarantee settlement. If the unhedged maker buys a position believed to win with probability 0.65, the full 35% wrong-way resolution risk remains until the contract resolves or the position is actually closed.