Probability Sampling

A sampling design is a probability sample if every unit $i$ in the population has a known, nonzero inclusion probability ${\pi }_i>0$, determined entirely by the design and not by the units themselves; simple random sampling, stratified sampling, cluster sampling, systematic sampling, and multistage sampling with probability-proportional-to-size selection are all probability sampling designs, differing only in how ${\pi }_i$ is structured. Because ${\pi }_i$ is known for every sampled unit, design-unbiased estimators of population quantities exist (most generally the Horvitz-Thompson estimator, which weights each observation by $1/{\pi }_i$), and the sampling variance of those estimators can be validly computed directly from the design, this is what allows probability samples to support formal statistical inference, confidence intervals and hypothesis tests with a rigorous frequentist justification, in a way non-probability samples cannot.

• Knowing π_i for every unit is what makes design-based inference possible at all, the entire apparatus of confidence intervals, standard errors, and margins of error reported by credible surveys rests on this single requirement, which is why 'margin of error ±3%' claims attached to non-probability samples (like unweighted opt-in web polls) are generally not statistically defensible in the same way.
• Probability sampling does not require equal probabilities across units, unequal-probability designs like probability-proportional-to-size sampling are still valid probability samples as long as every π_i is known in advance and used correctly in estimation (typically via Horvitz-Thompson-style weighting).
• A probability sample can still suffer from nonresponse bias, coverage error, or other non-sampling errors even when the design itself is sound, probability sampling guarantees that sampling error is quantifiable, it does not by itself guarantee the sample is free of other sources of error.