Sampling means choosing items from a population to collect data or model chance. When sampling with replacement, each chosen item is put back before the next draw, so the population stays the same. When sampling without replacement, chosen items are not returned, so the population changes after each draw.
This difference matters because it controls whether probabilities stay constant or change from trial to trial.
Understanding Statistics: Sampling With and Without Replacement
A small example makes the changing odds easier to see. Imagine a bag with five red counters and three blue counters. If the first counter is red and stays out of the bag, four of the seven remaining counters are red.
If the first counter is blue, five of the seven remaining counters are red. The result of the first draw therefore gives information about the second draw. This is called conditional probability.
For a particular sequence, such as red then blue, find the chance of the first result, then multiply by the chance of the next result after the first result is known. Keep track of the new total each time.
Replacement creates a different kind of repeated experiment. A spinner is effectively reset after every spin, and a shuffled deck that is fully restored before each draw behaves similarly. The chance of a result on one draw gives no clue about the next result.
This makes counting outcomes more straightforward. If a student draws four times and wants exactly two red counters, the two red results can occur in several different positions. Red red blue blue is one order.
Red blue red blue is another. Each valid order has the same probability when the chance stays fixed, so the number of possible orders matters as well as the probability of one order.
For draws from a finite group without replacement, it is often clearer to count complete samples rather than follow every possible order. Suppose a club has twelve members, including seven who play an instrument, and a committee of three is selected. A sample with two musicians can be counted by choosing two from the seven musicians and one from the five non-musicians.
That count is compared with the number of all possible committees of three. This method avoids accidentally counting the same committee many times in different orders. An important effect appears here.
Draws without replacement tend to be less spread out than draws with replacement. After several musicians have been chosen, fewer remain available, which limits extreme results.
Students meet these ideas in card games, raffle tickets, quality checks, surveys, and laboratory work. A card dealt from a hand is normally not returned. A computer simulation may return each selected item so that thousands of trials can be run under identical conditions.
In surveys, sampling without replacement is common because one person should not be selected twice. Random selection still matters. A sample can be without replacement yet biased if only certain people had a chance to be chosen.
When choosing a model, first identify the population size, the sample size, and whether items can reappear. If the sample is tiny compared with the population, treating the chance as nearly constant can be reasonable.
A common classroom guideline is that the sample should be no more than about ten percent of the population. State this approximation clearly rather than treating it as exact.
Key Facts
- With replacement: each draw has the same probability, so trials are independent.
- Without replacement: probabilities usually change after each draw, so trials are dependent.
- Multiplication rule for a sequence: P(A and B) = P(A)P(B given A).
- Binomial model with replacement: P(X = k) = C(n,k)p^k(1 - p)^(n - k).
- Hypergeometric model without replacement: P(X = k) = C(K,k)C(N - K,n - k)/C(N,n).
- Without replacement, if the sample is small compared with the population, the binomial model can be a close approximation.
Vocabulary
- Sample
- A sample is a group of items selected from a larger population.
- Replacement
- Replacement means returning a selected item to the population before making the next selection.
- Independence
- Events are independent if the outcome of one event does not change the probability of another event.
- Binomial distribution
- A binomial distribution models the number of successes in a fixed number of independent trials with the same success probability.
- Hypergeometric distribution
- A hypergeometric distribution models the number of successes in a fixed number of draws without replacement from a finite population.
Common Mistakes to Avoid
- Using the same denominator after drawing without replacement. This is wrong because the total number of items decreases after each draw.
- Treating without-replacement draws as independent. This is wrong because removing one item changes what is left for the next draw.
- Using the binomial formula when sampling a large fraction of a small population without replacement. This is wrong because the success probability is not constant across draws.
- Forgetting combinations in unordered sampling problems. This is wrong because many probability questions count groups, not ordered sequences.
Practice Questions
- 1 An urn has 5 red balls and 7 blue balls. If two balls are drawn with replacement, what is the probability that both are red?
- 2 An urn has 5 red balls and 7 blue balls. If two balls are drawn without replacement, what is the probability that both are red?
- 3 A factory has 1000 batteries, of which 20 are defective. A tester checks 5 batteries without replacement. Explain why a binomial model might still give a good approximation, and state what the success probability would be.