Estimating Probabilities Accurately

Every quantum measurement yields a single classical outcome — you never directly observe a probability. Instead, you run the same circuit many times (called shots) and count how often each outcome appears. The fraction of runs that produced outcome $b$ is your estimate $\hat{p}_b$ of the true probability $p_b = |\langle b|\psi\rangle|^2$ given by the Born rule.

From counts to error bars

Because each shot is an independent draw from a Bernoulli distribution with parameter $p$, the standard error of the proportion estimator over $N$ shots is

This is the typical deviation you expect between your estimate $\hat{p}$ and the true value $p$. The worst case is $p = 0.5$ (maximum uncertainty), where

To cut the error in half you need four times as many shots: precision scales as $1/\sqrt{N}$. This square-root law governs any sampling experiment, classical or quantum.

Practical consequences

When you run a quantum circuit on real hardware or a simulator, the number of shots sets a hard floor on how precisely you can know any output probability. For a lesson's grader based on analytic probabilities (exact formulas), this limitation disappears — but for real devices it is unavoidable. As a rule of thumb, choose $N$ large enough that $\mathrm{SE} \ll$ the smallest probability difference you need to resolve.

Try it

Solve for the minimum number of shots $N$ that guarantees $\mathrm{SE} \leq 0.01$ for any 50/50 distribution, then return that integer.

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