Direct answer
Sampling with replacement preserves the population composition between draws and can support independent identical trials; sampling without replacement changes the remaining composition and makes later draw probabilities conditional.
Visual explanation
Replacement determines the next sample space
What this calculation tells you
Replacement determines whether one selection changes the opportunity set for the next. The distinction controls branch probabilities and often determines whether binomial or hypergeometric models are appropriate.
Write the population counts before each draw. If selected items leave the population, update both favorable and total remaining counts; if they return, verify that the replacement and mixing mechanism restores the original probabilities.
Where it is used
Quality sampling
Model inspected units retained or returned to a finite lot.
Card and urn problems
Make changing sample spaces visible.
Auditing
Track finite selections from a known population.
Survey design
Introduce finite-population dependence while keeping real survey complexity separate.
Common situations
- Drawing several items from a finite lot.
- Choosing binomial or hypergeometric probability.
- Building a conditional tree for sequential draws.
- Explaining why later probabilities change.
Start with the statistical question
Write the population counts before each draw. If selected items leave the population, update both favorable and total remaining counts; if they return, verify that the replacement and mixing mechanism restores the original probabilities.
Two urn sequences begin identically. In one, the selected ball returns and the next urn is unchanged; in the other, the ball remains outside and the next branch shows updated counts.
Worked example
Drawing two red balls from 3 red and 2 blue gives (3/5)(3/5)=0.36 with replacement but (3/5)(2/4)=0.30 without replacement.
Assumptions that carry the result
Items must be distinguishable by the modeled category, draws must follow the stated selection mechanism, and replacement must genuinely restore the relevant population state.
Interpret the result without overreaching
The formulas do not validate random mixing, equal selection probabilities, hidden stratification, nonresponse, or physical process effects. Those can create dependence beyond replacement status.
- Keeping the denominator fixed without replacement.
- Updating the numerator but not the total remaining count.
- Using a binomial model for a large sampling fraction without considering dependence.
Practical questions
Frequently asked questions
Are draws with replacement always independent?
They can be under a stable random mechanism, but replacement alone does not eliminate poor mixing or other shared influences.
When is the binomial approximation reasonable?
It can be useful when the sampling fraction is small and assumptions are suitable, but the exact finite-population model remains distinct.
Does order matter without replacement?
Path probabilities depend on order, while some final count events combine several ordered paths into one unordered result.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.