Exercise: Probability Zero Is Not Impossible

Prerequisites: Probability Space

Problem

One of the most persistent misconceptions in probability is conflating "probability zero" with "impossible."

1. Let $U \sim \text{Uniform}[0, 1]$, defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with $\Omega = [0, 1]$, $\mathcal{F} = \mathcal{B}([0,1])$, and $\mathbb{P}$ Lebesgue measure. Compute $\mathbb{P}(U = 0.5)$. Is the event $\{U = 0.5\}$ impossible?

2. More generally, for any $x \in [0, 1]$, compute $\mathbb{P}(U = x)$. What does this say about individual outcomes in a continuous probability space?

3. In a continuous stock price model, $\mathbb{P}(S_T = K) = 0$ for every fixed $K$. A trader argues: "The probability of expiring exactly at-the-money is zero, so I don't need to worry about it." Identify the flaw in this reasoning. What concept should the trader use instead?

4. Let $A$ be any event with $\mathbb{P}(A) = 0$. Must $A = \emptyset$? Let $B$ be any event with $\mathbb{P}(B) = 1$. Must $B = \Omega$?

Hint

For part 1, recall that $\mathbb{P}([a, b]) = b - a$ for Lebesgue measure. For part 3, think about what quantity actually matters for option pricing near the strike.