Why a Large Population Can Make a Difference
In Independent Versus Dependent Events in Context, you saw that removing an item can change the probability of what happens on a later draw. So draws without replacement are generally dependent: the population available for the next draw changes. But if the sample is small compared with the population, those changes may be tiny. In that setting, treating draws as approximately independent can be useful.
The 10% condition is a rule of thumb for deciding when this approximation is reasonable. It does not make the draws exactly independent, and it does not replace random selection. It helps us judge whether removing a relatively small sample is likely to change the relevant probabilities much.
Here, \(N\) is the number of individuals or items in the population, and \(n\) is the number selected for the sample. Compare the whole sample size with the population size, not just the number of draws mentioned in one part of a question. If a survey samples 60 people from a population of 900, for example, check \(60\) against \(0.10(900)=90\).
What Changes—and What the Condition Allows
Suppose \(K\) of the \(N\) items have a particular feature. The probability that the first item has the feature is \(K/N\). If that item is kept and has the feature, then \(K-1\) such items remain among \(N-1\) items. The probability that the next item has the feature, given that the first did, is \((K-1)/(N-1)\). Those probabilities are generally different, which is why the draws are not exactly independent.
This update follows the counting logic from Sampling Without Replacement and Conditional Probability: remove the items already selected and use the remaining counts. The 10% condition does not change that exact calculation. Instead, it indicates when the changing counts are small relative to the original population, so probabilities across the sample are often close enough for an independence approximation.
A useful way to think about the condition is that even after selecting the sample, at least 90% of the original population remains. The pool has changed, but not by much relative to its starting size. The condition is a guideline, not a mathematical boundary at which the situation suddenly switches from dependent to independent. A sample just above 10% is still dependent; it simply does not meet this guideline for using the approximation.
- The sample is selected randomly from a defined population.
- Sampling is without replacement, so the draws are not exactly independent.
- The sample size satisfies \(n\leq 0.10N\).
- The approximation is appropriate for the probability calculation at hand; if an exact calculation is manageable or needed, use the changing counts instead.
Random selection and the 10% condition address different concerns. Random selection supports treating the sample as representative of the population; the 10% condition supports treating draws within that sample as approximately independent. Meeting the 10% condition does not fix a biased selection method, and random selection does not make draws without replacement exactly independent.
A Practical Check
Before using an independence approximation for a sample taken without replacement, identify the population size and the intended sample size. Then calculate 10% of the population and compare the sample size with that amount. If the sample is no more than 10% of the population, you can usually use the original proportion for each draw as an approximation. If not, track how the counts change.
Use the full population size and the total number selected for the sample.
Calculate \(0.10N\), then check whether \(n\leq 0.10N\).
If the condition holds, an independence approximation may be reasonable. If it does not, use conditional probabilities with the remaining counts when calculating exactly.
Say “approximately independent” when using the condition. Do not claim that sampling without replacement is exactly independent.
Worked Example: A Small Sample of Residents
Worked Example: A Small Sample of Residents
A town has 2,000 residents, and 500 have a particular professional certification. A random sample of 80 residents is selected without replacement. Let \(A\) mean the first selected resident has the certification, and \(B\) mean the second selected resident has it. Check the 10% condition, then compare the exact probability of both events with an independence approximation.
State. We want to know whether it is reasonable to treat the draws as approximately independent for this sample, and how close that approximation is for the first two selections.
Plan. Compare the full sample size with 10% of the population. Then calculate the second-draw probability both overall and given that the first resident had the certification. As in The General Multiplication Rule, multiply the first-draw probability by the relevant conditional probability to find the probability of both events.
Do. The population size is \(N=2{,}000\), so 10% of the population is \(0.10(2{,}000)=200\). The sample size is \(n=80\), and \(80\leq200\), so the 10% condition is satisfied. The draws are still dependent in the exact process, but an independence approximation is reasonable.
Before any draw, the probability that the second resident has the certification is \(P(B)=500/2{,}000=0.25\). Given that the first resident had it, 499 certified residents remain among 1,999 residents. Thus,
The exact conditional probability is slightly below \(0.25\), as expected after one certified resident is removed. The exact probability that both selected residents have the certification is
If we instead treat the draws as independent, the approximation is \(P(A)P(B)=0.25(0.25)=0.0625\). As a check, the exact probability can also be calculated by counting ordered pairs: \(500(499)/[2{,}000(1{,}999)]\approx0.0624\). Both exact calculations agree to four decimal places.
Conclude. The 80-person sample is no more than 10% of the 2,000-person population, so treating the draws as approximately independent is reasonable. For the first two selections, the exact probability that both residents have the certification is about \(0.0624\), close to the independence approximation of \(0.0625\). The draws are not exactly independent because the first selection changes the remaining counts.
Worked Example: A Sample That Is Too Large for the Guideline
Worked Example: A Sample That Is Too Large for the Guideline
A warehouse has 80 identical-looking packages, and 24 are marked for inspection. A random sample of 20 packages is selected without replacement. Let \(A\) mean the first package is marked, and \(B\) mean the second package is marked. Check the 10% condition and compare the exact and independence calculations for both packages being marked.
State. We will decide whether the 10% condition supports an independence approximation and calculate the exact probability to see the effect of updating the counts.
Plan. Compare 20 with 10% of 80. If the guideline is not met, use the remaining number of marked packages and total packages after the first draw, rather than assuming that the second-draw probability stays unchanged.
Do. Ten percent of the population is \(0.10(80)=8\). Since the sample size \(20\) is greater than 8, the 10% condition is not satisfied. The initial probability of a marked package is \(24/80=0.30\). Given that the first package is marked, the probability the second is marked is \(23/79\approx0.2911\). Therefore, the exact probability that both are marked is
The independence calculation would be \(0.30(0.30)=0.0900\), which is not the exact answer. A separate check using ordered pairs gives \(24(23)/[80(79)]=552/6{,}320\approx0.0873\), the same result as the conditional calculation.
Conclude. The sample is more than 10% of the population, so this guideline does not support treating the draws as approximately independent. For the event that both packages are marked, using the exact updated probability gives about \(0.0873\); multiplying the original probability by itself gives \(0.0900\). The first marked package reduces the number of marked packages available for the second draw.
Failing the 10% condition does not automatically mean an independence approximation will be inaccurate for every possible question. It means this particular rule of thumb does not support that approximation. When the sample is a noticeable share of the population, updating the counts is often straightforward and avoids relying on an approximation.
Worked Example: Three Items from a Large Quality-Control Lot
Worked Example: Three Items from a Large Quality-Control Lot
A production lot contains 5,000 items, of which 1,250 meet a specified quality standard. An inspector randomly selects 100 items without replacement. Find the exact probability that the first three selected items all meet the standard, then compare it with the independence approximation.
State. We want the probability of three consecutive selections with the specified feature. The sample size, not just the three draws in the event, determines whether the 10% condition is met.
Plan. Check the full sample of 100 against the population of 5,000. Then calculate the exact probability using the updated counts after each successful selection. Compare that result with multiplying the original feature proportion three times.
Do. Ten percent of the population is \(0.10(5{,}000)=500\), and \(100\leq500\). The 10% condition is satisfied, so an independence approximation is reasonable. The original proportion meeting the standard is \(1{,}250/5{,}000=0.25\). For the exact calculation, after one qualifying item is selected, 1,249 of 4,999 items qualify; after a second, 1,248 of 4,998 qualify. Thus,
More precisely, the exact probability is approximately \(0.0155969\), rounded to seven decimal places. Treating the draws as independent gives \(0.25^3=0.015625\), or approximately \(0.0156\) to four decimal places. The two answers are close; using more digits shows the small difference.
Conclude. Because the sample is only 2% of the lot, the 10% condition is satisfied, and the independence approximation is reasonable. The exact probability that the first three selected items all meet the standard is approximately \(0.0155969\); the independence approximation is \(0.015625\). The exact calculation remains a little smaller because each qualifying selection slightly reduces the qualifying proportion among the items left.
Common Mistakes and AP Exam Tips
- Calling the draws exactly independent. The 10% condition supports an approximation; it does not erase the effect of removing items. Say “approximately independent” or “reasonable to treat as independent.”
- Checking only the number of events in the calculation. If a sample of 100 is drawn but a question asks about the first two draws, check \(n=100\), not just 2, against 10% of the population.
- Confusing the population and sample sizes. Use \(N\) for the full population and \(n\) for the sample. The condition is \(n\leq0.10N\), not \(N\leq0.10n\).
- Assuming a failed condition proves the approximation is terrible. It means this guideline does not justify using the approximation. Compare exact values when possible, or calculate with the updated counts.
- Forgetting that the 10% condition is not a randomness check. A large population does not make a convenience sample representative. Explain how the sample was selected separately.
- Leaving out the updated denominator. After one item is removed, the total remaining is \(N-1\), not \(N\). If the item had the feature, the feature count also decreases by one.
Key Takeaway
Sampling without replacement changes the remaining population, so its draws are not exactly independent. When the sample is no more than 10% of the population, that change is often small enough to use an independence approximation. When the condition is not met, or exact probabilities are needed, use the changing counts and conditional probabilities.
Check Your Understanding
For each situation, decide whether the 10% condition is satisfied and explain what that does—or does not—allow you to conclude.
- A random sample of 45 residents is selected without replacement from a population of 600. Is the 10% condition satisfied?
- A factory samples 30 items without replacement from a lot of 250. What is 10% of the lot, and does the sample meet the condition?
- In your own words, why are draws without replacement not exactly independent even when the 10% condition is satisfied?
- A sample is no more than 10% of the population, but it was selected by asking only volunteers to respond. Does the 10% condition make the sample representative? Explain.
- A sample exceeds 10% of its population. What calculation can you use to find an exact probability for successive draws?