With vs without replacement sampling

With vs without replacement: the short answer

The difference is whether you put each pick back before the next one. With replacement, you return the item, so the pool never changes, every draw faces the same probabilities, and the draws are independent. Without replacement, you keep each item out, so the pool shrinks, the probabilities shift on every draw, and the draws are dependent.

That one choice decides how you multiply. Independent draws let you multiply the same probability each time, while dependent draws force you to update the probability using what has already been removed.

Sampling with replacement

With replacement, an item selected from the population is returned before the next selection, so it can be chosen again. Because the pool is restored each time, the probability of any outcome stays the same on every draw, and the draws are independent.

That independence makes the arithmetic simple. For two draws, $P(A \text{ then } B) = P(A) \cdot P(B)$, using the same probabilities each time. This is the setup behind the binomial distribution, which requires a fixed probability of success across independent trials. Flipping a coin or rolling a die behaves like sampling with replacement, since nothing is used up.

Sampling without replacement

Without replacement, an item is selected only once and is not returned, so the pool gets smaller with each draw. The probabilities change from draw to draw, which makes the draws dependent: what you pick first affects what is left for the second.

Here you use the general multiplication rule with a conditional probability: $P(A \text{ then } B) = P(A) \cdot P(B \mid A)$, where $P(B \mid A)$ reflects the smaller, updated pool. Dealing cards, drawing names from a hat, or selecting people for a committee all work this way, because once chosen, an item is gone. Most real sampling is done without replacement, so no one is surveyed twice.

With vs without replacement side by side

When to use which, and the 10% shortcut

Use with replacement when items truly can repeat, or when a model such as the binomial assumes independent trials with a constant probability. Use without replacement for most real sampling, where each person or object is selected at most once.

There is a useful bridge between them. When you sample without replacement but the sample is small relative to the population, the probabilities barely move, so the draws are close enough to independent to treat with the simpler with-replacement math. The AP course formalizes this as the 10% condition: when sampling without replacement, if the sample size $n$ is at most 10% of the population size $N$, you may treat the draws as approximately independent.

The classic mix-up and how to avoid it

The classic error is multiplying the same probability twice for a without-replacement problem, which ignores that the pool shrank. Drawing two red marbles is not $\frac{4}{10} \times \frac{4}{10}$ when you do not replace the first; the second draw must use the smaller pool.

Two habits prevent it. First, read whether the item is returned: words like 'replace,' 'put back,' or 'with replacement' signal constant probabilities, while 'without replacement,' 'do not return,' or dealing and selecting people signal a shrinking pool. Second, for without-replacement draws, update both the count of successes and the total before the next fraction, so the denominator drops by one on each successive draw.

Where this fits in AP Statistics

The two sampling strategies are defined in Unit 1 topic 1.11, and the probability machinery lives in Unit 2: independent events and the multiplication rule in topics 2.6 and 2.7. The binomial probability model, computed with the binomial probability calculator, assumes independent trials, which is the with-replacement setup.

The 10% condition shows up again in inference, where sampling without replacement is treated as approximately independent as long as the sample is at most 10% of the population. Keeping disjoint separate from independent events helps here, since replacement is about independence across draws, not about whether two events can co-occur.