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.
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.