Sampling without replacement

Sampling without replacement is a selection mode where the item chosen is not put back into the set. Once it comes out, it is set aside and cannot return. Each item can therefore be drawn only once, which sharply distinguishes this mode from sampling with replacement.

The simplest image is that of an urn from which you remove balls without ever returning them. With each draw, the urn contains one ball fewer, like a bag that empties little by little until there is nothing left to draw.

Two consequences result. On one hand, the set shrinks and the probabilities change with each removal. If an urn contains ten balls, the first has one chance in ten of coming out; once a ball has been removed, nine remain, and the next is played on one chance in nine. On the other hand, the draws are dependent: what has already been removed changes what is left, so each result influences the chances of later draws.

A common misunderstanding is to reason as if the chances stayed fixed. In reality, the further the draw advances, the more each item still present sees its probability of being chosen increase, simply because fewer and fewer competitors remain in the set. Ignoring this shrinking leads to wrong estimates.

On the site, this mode corresponds to all draws where you want to avoid duplicates: designating several distinct winners, drawing the numbers of a lottery where a number already drawn cannot come back, or splitting participants into teams without the same person being placed twice. Whenever a result must be unique, sampling without replacement is what applies.