Probability Primer
Random variables and distributions — what measurement outcomes actually are
By the end of this topic you'll be able to
Probability enters physics in two distinct ways, Binney and Skinner point out: through data we simply haven't measured precisely enough, and through systems genuinely subject to random influences. A pendulum's period is uncertain only because we haven't characterized its length precisely; a thermos of water left on a table is uncertain because of draughts we could, in principle, track down and eliminate.
When events are repeatable — a die thrown again and again — we assign a probability pᵢ to each outcome so that, over N trials, the number of times outcome i occurs is approximately pᵢN, with the approximation improving as N grows. With N mutually exclusive outcomes, the probabilities must sum to exactly 1; if no outcome is favored over any other, each must equal 1/N.
A random variable is a quantity whose value is uncertain until measured — the number that comes up on a die, say. Its expectation value ⟨x⟩ = Σᵢ pᵢxᵢ is the average value obtained over many trials, and for two random variables x and y, ⟨x+y⟩ = ⟨x⟩+⟨y⟩ regardless of whether the variables are independent.
No flips yet — each flip is p(heads) = p(tails) = 1/2, but any small run can look lopsided.
Flip just a few times and the split is often far from 50/50. Flip a thousand and it settles in close to it — the probability pₐ is a property of the long run, not a promise about any one trial.
Bayes' theorem lets this framework update itself as new evidence arrives: P(A|B) = P(B|A)P(A)/P(B). It's the same rule an error-correction decoder uses to infer, from a handful of syndrome measurements, what most likely went wrong inside a much larger system.
Here is the point Binney and Skinner press hardest, and the one that matters most for everything ahead: in the classical cases above, uncertainty could in principle always be reduced by more careful work — isolate the thermos better, characterize the pendulum more precisely, and the randomness shrinks toward zero.
Quantum mechanics breaks this pattern entirely. Even given the most complete information about a system's state that it is logically possible to have, the outcomes of some measurements remain uncertain — not because our knowledge is incomplete, but because, as far as the theory and a century of experiments can tell, there is no further fact to know. Whereas classical uncertainty can be made as small as we please by careful work, in the quantum world uncertainty is woven into the fabric of reality itself.
Try It Yourself
A qubit-readout circuit has a 95% chance of correctly reporting a qubit's true state, and a 10% chance of a false '1' reading when the true state was 0. If a qubit is prepared in |0⟩ with certainty and read out as '1', use Bayes' theorem to find how likely it is the readout was simply wrong (rather than the qubit somehow having genuinely been 1).
- 1Since the qubit was prepared in |0⟩ with certainty, P(true state = 0) = 1 and P(true state = 1) = 0 going in — this is a readout-error question, not a genuine superposition question.
- 2We're told P(reads '1' | true 0) = 0.10 (the false-positive rate) and P(reads '1' | true 1) = 0.95 (the true-positive rate).
- 3Apply Bayes' theorem: P(true 0 | reads '1') = P(reads '1' | true 0)·P(true 0) / P(reads '1').
- 4Since P(true 1) = 0 here, P(reads '1') = P(reads '1'|true 0)·P(true 0) + P(reads '1'|true 1)·P(true 1) = 0.10·1 + 0.95·0 = 0.10.
- 5So P(true 0 | reads '1') = (0.10 · 1) / 0.10 = 1.
Probability 1 — since the qubit was prepared in |0⟩ with certainty, any '1' reading must be a readout error, not evidence the qubit was genuinely 1. (Change the setup so the prepared state itself is uncertain, and Bayes' theorem gives a genuinely informative answer instead of this degenerate case.)
Reference
| Random variable | A quantity with an uncertain value until observed | |
| Probability distribution | Likelihood assigned to each possible outcome | |
| Expectation value | The average outcome, weighted by probability | |
| Bayes' theorem | Updates a probability given new evidence |
Quick Check
What must be true of any valid probability distribution over outcomes?
What's the key conceptual difference between a classical coin flip and a qubit in superposition?