Sampling with replacement

Sampling with replacement is a selection mode where each item chosen is put back into the set before the next draw. The set therefore stays unchanged from one draw to the next, and the same item may well come up several times in a row. This is the very definition of a draw that never "consumes" what it takes.

The clearest image is that of an urn in which you draw a ball, note its color, then put it back before reaching in again. With each reach, the urn is in exactly the same state as at the start: nothing keeps a trace of the previous draws.

Two important consequences follow. First, the probabilities stay constant: if an urn contains four balls, the chance of drawing a given ball remains one in four every time. Second, the draws are independent: the previous result has no influence on the next. Rolling a die three times is the textbook example, because the die does not remember its past rolls and a "six" can come up on all three throws.

This is where a famous trap slips in. After several identical results, many people feel that "the other outcome is due" and is bound to happen eventually. This intuition is false in sampling with replacement: each draw starts from scratch, and no compensation is held in reserve. The chances do not rebalance on their own from one draw to the next.

On the site, this mode corresponds to anything that must be able to repeat identically: rolling a die again, flipping a coin again, or drawing again from a single pool where every outcome stays available. You choose it whenever you want each draw to be "fresh" rather than gradually exhausting the possibilities.