Sample Spaces and Events

Probability is the language of quantum mechanics. Every measurement of a quantum system ends in a definite classical result — a spin up or down, a photon detected or not — but quantum theory only predicts the likelihood of each possibility. Before we can make sense of those likelihoods we need a precise vocabulary: outcomes, events, and probability.

The sample space

A sample space $\Omega$ is the set of all possible outcomes of an experiment. Each element $\omega \in \Omega$ is a single, indivisible result. A few examples:

Flipping a fair coin: $\Omega = \{\text{H}, \text{T}\}$ .
Rolling a six-sided die: $\Omega = \{1, 2, 3, 4, 5, 6\}$ .
Measuring the spin of an electron along the $z$ -axis: $\Omega = \{+\tfrac{\hbar}{2}, -\tfrac{\hbar}{2}\}$ , often written $\{+, -\}$ or $\{|0\rangle, |1\rangle\}$ .

The last example is the one we will return to throughout this course. A spin measurement has exactly two outcomes, so its sample space has exactly two elements — it is the simplest quantum sample space imaginable.

Events

An event $A$ is any subset of the sample space, $A \subseteq \Omega$ . Events let us group outcomes and ask "did something in this group happen?"

On the die, "rolling an even number" is the event $A = \{2, 4, 6\}$ . On the spin measurement, "getting spin-up" is the event $A = \{+\tfrac{\hbar}{2}\}$ , which happens to be a single-element set (a so-called elementary event). The empty set $\emptyset$ is the event that never happens; $\Omega$ itself is the event that always happens.

Probability

A probability function assigns a number $P(A) \in [0, 1]$ to every event $A$ in a way that satisfies three axioms (stated formally in the next lesson). For now the key intuition is:

$P(\emptyset) = 0$ — the impossible event has probability zero.
$P(\Omega) = 1$ — something always happens.
If $A$ and $B$ share no outcomes, $P(A \cup B) = P(A) + P(B)$ .

For a fair coin each elementary event gets probability $\tfrac{1}{2}$ . For a fair die each gets $\tfrac{1}{6}$ , so the probability of rolling an even number is $P(\{2,4,6\}) = 3 \times \tfrac{1}{6} = \tfrac{1}{2}$ .

Why this matters for quantum mechanics

In quantum mechanics, measurement outcomes come with probability amplitudes — complex numbers $\alpha_i$ whose squared moduli give probabilities: $P(\omega_i) = |\alpha_i|^2$ . The sample space is still the set of possible measurement outcomes; the innovation is the rule for computing each $P(\omega_i)$ . Understanding $P(\omega_i)$ as a plain probability function first makes the quantum rule cleaner when it arrives.