Sampling With and Without Replacement
The single decision that changes every number downstream of your draw.
Every repeated draw makes one decision before anything else: does the drawn item go back into the pool? That single choice changes the odds, the variance, whether repeats are possible, and what your result can support.
Putting it back is called sampling with replacement. Keeping it out is sampling without replacement. Neither is more correct — they model different situations, and the mistake is using one where the other belongs.
This page covers what changes, how to tell which you need, and the specific arithmetic that trips people up.
What actually changes
With replacement, every draw is independent and identical. The pool is the same size every time, the odds never change, and the same item can come up repeatedly. Ten draws from a hundred items are ten separate 1-in-100 events.
Without replacement, every draw shrinks the pool. The first is 1-in-100, the second is 1-in-99, the tenth is 1-in-91. Draws are no longer independent — knowing what came out first tells you something about what can come out next — and repeats are impossible.
The most visible consequence is coverage. Ten draws with replacement from a hundred items will typically produce nine or ten distinct items, and occasionally eight. Ten draws without replacement always produce exactly ten.
10 draws from a pool of 100
- With replacement: every draw 1-in-100; expected distinct items about 9.6.
- Without replacement: draws run 1-in-100 down to 1-in-91; distinct items exactly 10.
- The chance of at least one repeat with replacement is about 37%.
Which one you need
The test is whether drawing an item consumes it. If you are physically going to inspect a document, interview a person, or award a prize, the item is consumed and you want without replacement — inspecting the same document twice tells you nothing new.
If each draw is an independent event that leaves the world unchanged, you want with replacement. Rolling a die, generating a password character, simulating a coin flip — nothing is used up, and forcing distinctness would introduce a dependency the model does not have.
The ambiguous cases are usually resolved by asking what a repeat would mean. If a repeat is meaningless or annoying, you want without replacement. If a repeat is a legitimate outcome that carries information, you want with.
- Consumed by drawing (audit, prize, interview): without replacement.
- Independent events (dice, passwords, simulations): with replacement.
- Repeats meaningless or annoying: without replacement.
- Repeats informative or legitimate: with replacement.
The variance difference
Sampling without replacement produces less variable results than sampling with, and the effect grows as your sample approaches the population size. The adjustment is called the finite population correction, and it is often ignored when it should not be.
Concretely: sampling 50 of 100 items without replacement gives an estimate with about 71% of the standard error you would get from 50 independent draws. Sampling 90 of 100 gives about 32%. Sampling all 100 gives zero error, because you have measured everything.
That is why a sampling fraction matters and not just a sample size. A sample of 50 means something quite different from a population of 100 than from a population of a million, and reporting only the sample size hides the difference.
The counting trap
The most common arithmetic error is computing per-draw odds and treating them as overall odds. In a raffle of 100 tickets drawing 5 winners without replacement, your chance of winning is not 1-in-100.
The correct figure is 5-in-100, or 5%. Each of the five draws is a separate opportunity, and since you cannot win twice, the chances add cleanly rather than requiring a more complicated calculation.
With replacement it is subtler, because you could win more than once. Five independent draws from 100 give you a 1 − (99/100)⁵ chance of winning at least once, which is about 4.9% — very slightly less than 5%, because some of the probability is spent on you winning twice.
Winning something in a 5-prize draw from 100 entrants
- Without replacement: exactly 5% — five draws, no repeats, chances add.
- With replacement: 1 − (0.99)⁵ ≈ 4.9% of winning at least once.
- The small gap is the probability of winning twice, which the first method excludes.
The hybrid: no-repeat cycles
Several tools on this site use a middle case worth naming: draw without replacement until the pool empties, then refill and start again. That is sampling without replacement within a cycle, and with replacement across cycles.
It gives a coverage guarantee — everyone is drawn once before anyone is drawn twice — while remaining usable indefinitely. The cost is that the odds within a cycle are not equal: the first draw of thirty names is 1-in-30 and the last is a certainty.
It also means a boundary effect: someone drawn at the end of one cycle can be drawn at the start of the next, producing an apparent repeat that the guarantee does not cover. That is the honest limitation, and the tools that use it say so.
Frequently asked questions
What is the difference in one sentence?
With replacement, the drawn item goes back and the odds never change; without replacement, it stays out and the pool shrinks with every draw.
How do I know which one I need?
Ask whether drawing an item consumes it. An audit or a prize draw consumes; a dice roll or a password character does not.
Does it change the overall odds of being picked?
For k draws from n without replacement, your chance is exactly k/n. With replacement it is 1 − ((n−1)/n)^k, which is slightly lower.
Why is the with-replacement figure lower?
Because some of the probability is spent on you being drawn twice, which does not add to your chance of winning at least once.
What is the finite population correction?
The reduction in variance from sampling without replacement. Taking 50 of 100 gives about 71% of the standard error of 50 independent draws.
Why does the sampling fraction matter?
Because 50 from a population of 100 is a very different measurement from 50 from a million, and reporting only the sample size hides that.
How many distinct items do I get with replacement?
Fewer than you drew. Ten draws from a hundred give about 9.6 distinct items on average, with at least one repeat about 37% of the time.
What is a no-repeat cycle?
Without replacement within a cycle, with replacement across cycles. It guarantees coverage while remaining usable indefinitely, at the cost of unequal odds within each cycle.
Tools that use this
Random Sample From a List
Draw a fixed-size sample without replacement — for audits, spot checks, and surveys.
Random Subset Picker
Include each entry independently at probability p — the sample size is a random variable.
Bingo Caller
Call numbers without replacement, track the board, and verify a winning card.
Related guides
Random Sampling for Audits and Surveys
Sampling frames, sample size, and the mistake that invalidates the whole exercise.
How Weighted Random Selection Works
Cumulative sums, one float, and why duplicating entries is the wrong way to do it.
The Birthday Problem and Collisions
Why 23 people is enough for a shared birthday, and where else the same maths ambushes you.