The 10% Condition for Independence in Binomial Settings

Why Sample Size Matters When Sampling Without Replacement

In Recognizing a Binomial Setting and The BINS Checklist for Binomial Conditions, you learned that independent trials and a constant probability of success are part of a binomial setting. But when you sample people or objects from a finite population without replacement, the selections are not truly independent: after each selection, the population left to sample from has changed.

The 10% condition gives a practical rule for deciding when this dependence is small enough to ignore for a binomial model. It applies to a random sample taken without replacement from a population whose size is known. The condition does not make the selections independent; it supports treating them as approximately independent.

What Changes After Each Selection?

Suppose a population contains \(N\) individuals, of whom \(K\) have the characteristic labeled success. Before sampling, the probability that a randomly selected individual is a success is \(K/N\). If sampling is without replacement, that probability can change after the first selection. For example, after selecting a success, there is one fewer success in the population; after selecting a failure, there is one fewer failure.

Here is a small illustration. Imagine a population of 1,000 items, 200 of which are successes. The first selection has success probability \(200/1000=0.20\). After selecting one success, the next success probability is \(199/999\approx0.1992\). After selecting one failure, the next success probability is \(200/999\approx0.2002\). These probabilities are not exactly the same, and the selections are not truly independent.

If the sample is small relative to the population, however, each selection changes only a small part of the population. The success probability and the dependence between selections then change only slightly, which makes a binomial model a reasonable approximation in many settings. The 10% condition is a practical guideline for this purpose; it is not a claim that dependence disappears.

When the condition holds, the binomial model uses \(n\) as the number of sampled individuals and \(p=K/N\) as the population proportion of successes. The random variable \(X\) counts the successes in the sample, so the approximate model is \(X\sim B(n,p)\). As discussed in Defining \(n\), \(p\), and \(X\) in Context, define the success and the count in the situation before using this notation.

The 10% condition addresses the independence part of BINS and helps support the same-probability part. It does not replace the other checks. You still need two possible outcomes for each trial, a fixed number of trials, and a random selection process that makes the stated probability model sensible. A sample that is not selected randomly can be biased, even if it is less than 10% of the population.

How to Check the Condition

First identify the population being sampled and count its members to get \(N\). Then identify the number selected, \(n\). Multiply the sample size by 10 and compare that result with the population size. If \(N\geq10n\), the 10% condition is satisfied. If \(N<10n\), it is not.

You can also calculate the sample’s fraction of the population, \(n/N\), and check whether it is at most 0.10. These two methods are equivalent. For example, if \(n=40\), the population must have at least \(10(40)=400\) members. A population of 400 meets the condition exactly; a population of 399 does not.

Be precise about what counts as the population. If a survey samples from eligible registered participants, \(N\) is the number of eligible participants in that sampling frame—not everyone who lives in the area or belongs to a broader group. Likewise, use the sample size actually drawn, not the number who later respond, unless the problem specifically defines the sample that way.

Worked Example: Inspecting a Small Fraction of an Inventory

Worked Example: A Sample That Is Too Large Relative to the Population

Worked Example: Meeting the Condition Exactly

What the Condition Does—and Does Not—Tell You

The condition is a check on how much the sample can change the population as sampling proceeds. If a small fraction of the population is sampled, removing one selected member has a relatively small effect on the mix of successes and failures left. If a large fraction is sampled, the remaining mix can change more noticeably, so the selections are more strongly dependent.

Passing the 10% condition does not prove that a binomial model is perfect or guarantee a particular level of accuracy for every probability. It is a practical AP Statistics guideline for treating sampling without replacement as approximately independent. The actual process is still without replacement, and so its selection outcomes remain dependent.

Also, the condition is not a substitute for checking that the sample was randomly selected. A convenience sample of 5% of a population can satisfy the numerical inequality while still failing to represent that population well. The condition addresses the effect of sampling without replacement; random selection addresses how the sample was obtained.

If the population size is not given or cannot be determined, you cannot verify \(N\geq10n\) from the available information. Do not assume that the condition holds just because the sample sounds small. State what additional population-size information is needed, or explain that the binomial approximation is not justified by the information provided.

Common Mistakes and AP Exam Tips

• Reversing the inequality. The condition is \(N\geq10n\), not \(n\geq10N\). Compare population size with ten times the sample size.
• Checking only whether the sample seems small. Use the stated population and sample sizes. For \(n=500\), for example, the required population size is at least \(5{,}000\).
• Calling the selections independent after the check passes. Sampling without replacement creates dependence. Say that the condition supports treating the selections as approximately independent.
• Assuming the 10% condition verifies all of BINS. It addresses dependence in a without-replacement sample. You must still identify two outcomes, a fixed number of selections, and a reasonable probability model.
• Using the wrong population size. Use the population from which the sample was actually drawn, such as all eligible members on a list—not a larger group that includes people who could not have been selected.
• Ignoring how the sample was selected. A small sample fraction does not make a nonrandom sample representative. Describe the selection process as well as checking the numerical condition.

For full-credit communication, identify \(N\) and \(n\), show the comparison \(N\geq10n\) or calculate \(n/N\), and state whether the condition is satisfied. Then connect that result to the binomial model in context. If sampling is without replacement, describe the model as an approximation rather than claiming the selections are truly independent.

Key Takeaway

The 10% condition connects finite-population sampling to the independence requirement in BINS. It gives a clear numerical check, but the conclusion is always about a reasonable approximation—not exact independence.