Why Check the Conditions?
A count such as “the number of patients who respond” or “the number of defective items in a sample” may look binomial, but a count is not automatically binomial just because it records successes. A binomial model describes a particular setup: a fixed number of trials, two possible outcomes on each trial, the same probability of success each time, and independent trial outcomes.
Checking these conditions matters before using binomial calculations. If the number of trials changes according to what happens, or if success probabilities vary, the model may not fit. If trials are dependent, the count’s probabilities may also differ from those given by a binomial model. Some real processes satisfy the conditions exactly; others meet them only approximately.
The Four Conditions
The first condition is a fixed number of trials. The value of \(n\) must be determined before observing the trial outcomes. For example, inspecting exactly 12 items gives a fixed number of trials. Inspecting items until the third defective one is found does not: the number inspected depends on the outcomes.
Second, each trial must have two outcomes for the purpose of the count. These are usually called success and failure. A trial may have more than two outcomes in its original form, but it can still be coded into two groups if the question counts whether a particular event occurred. For instance, a survey answer could have several categories; “selected option A” can be coded as success, and all other answers as failure. The coding must match the question being answered.
Third, the probability of success must be constant from trial to trial. If the chance changes because conditions change, a single value \(p\) does not describe every trial. A stable process may support a constant-probability model, but the assumption should be justified by the context rather than inferred from the fact that the same type of item or person is observed each time.
Fourth, the trials must be independent: knowing the outcomes of some trials should not change the probability distribution of the others. In particular, independence is stronger than saying that trials seem unrelated. A shared cause—such as a machine setting, household membership, or a change in testing conditions—can make outcomes dependent.
A quick memory aid is “fixed number, two outcomes, same probability, independent.” All four are needed for the usual binomial model. The phrase Bernoulli trial describes one trial with a success/failure outcome; a binomial count combines a fixed number of such trials under the additional constant-probability and independence conditions.
Worked Example: A Random Survey Without Replacement
A hypothetical city has 1,000 registered residents, of whom 240 support a proposed transit plan. A simple random sample of 50 residents is selected without replacement. Let success mean that a selected resident supports the plan. Is the number of supporters in the sample exactly binomial, approximately binomial, or neither?
Fixed number: The sample contains exactly \(n=50\) residents, so the number of trials is fixed.
Two outcomes: Each selected resident either supports the plan or does not, so each response can be coded as success or failure.
Constant probability: The probability on the first draw is \(240/1000=0.2400\). After one supporter has been selected, the probability on the next draw is \(239/999\approx0.2392\); after one non-supporter has been selected, it is \(240/999\approx0.2402\). Thus the probability is not exactly constant.
Independence: The draws are not exactly independent. For example, learning that the first resident supports the plan changes the chance that the second resident supports it from \(240/1000\) to \(239/999\). Sampling without replacement from a population containing a mix of outcomes generally makes trial outcomes dependent.
However, the sample is a small fraction of the population:
The sample is no more than 10% of the population. Under the common 10% guideline for sampling without replacement from a finite population, treating the draws as approximately independent is reasonable. The count is therefore approximately binomial, not exactly binomial. The 10% guideline is an approximation rule, not a claim that the draws become mathematically independent.
Why Sampling Without Replacement Usually Creates Dependence
The survey example illustrates a general fact. Suppose a population has \(N\) units, of which \(K\) are successes, and two distinct units are selected at random without replacement. If \(0<K<N\), the population contains a mix of successes and failures. The probability that both selected units are successes is
If the two outcomes were independent, this joint probability would instead equal the product of their individual success probabilities, \((K/N)^2\). Subtracting that product from the actual joint probability gives
where the inequality follows from \(0<K<N\) and \(N>1\). Therefore the two outcomes are not independent when the population contains both successes and failures. Selecting one success leaves one fewer success among the remaining units.
There is an important boundary case: if every unit in the population has the same outcome, then every draw has a constant outcome. Those outcomes are independent, even when selection is without replacement. Thus it is too broad to say that sampling without replacement always makes outcomes dependent. In ordinary surveys with a mix of responses, however, dependence is generally present; the 10% guideline addresses whether it is small enough to ignore for an approximation.
Checking the Conditions in Practice
A condition check should use details of how the data are generated. For repeated production checks, for example, an observed historical defect fraction can suggest a working value of \(p\), but it does not establish that the process will stay stable or that consecutive items are independent. Production changes, batches, or a malfunctioning machine can alter the probability or link outcomes.
Worked Example: Inspecting Items From a Stable Process
Suppose a hypothetical factory plans to inspect exactly 12 items from a production process. In earlier records, 18 of 600 items were defective. The quality team proposes using \(p=0.03\) as a working defect probability and assumes that the process remains stable and item outcomes are independent. Is the count of defective items reasonably modeled as binomial?
Fixed number: Exactly 12 items will be inspected, so \(n=12\) is fixed before the inspection.
Two outcomes: Each item is either defective (success, for this count) or not defective (failure).
Constant probability: The historical fraction is
This is an observed fraction, not proof that the true future defect probability is exactly 0.0300. If the production process is stable, it can serve as a reasonable working value for the common probability \(p\).
Independence: The model assumes one item's defect status does not affect another's. That assumption is plausible only if the items are produced under stable conditions without meaningful clustering or shared disruptions. If defects tend to occur in runs because of a machine problem, independence is doubtful.
With the stated stable-process and independence assumptions, the count is reasonably modeled as binomial with \(n=12\) and working success probability \(p=0.0300\). Without those assumptions, the historical fraction alone is not enough to justify the model.
A constant probability and independence are distinct conditions. In some settings each trial can have the same marginal success probability while the outcomes still depend on one another. Conversely, independence alone does not make trials binomial if their success probabilities differ. Check both rather than treating either as a substitute for the other.
Worked Example: A Count With a Stopping Rule
In a hypothetical screening exercise, staff continue testing people until the third positive result occurs. They record the number of positive results when testing stops. Assume each test has two outcomes, the same positive probability, and independent results. Is the recorded count binomial?
The count of positive results at the stopping time is always 3: that is the rule for ending the exercise. The number of tests, however, is not fixed in advance. One possible run might produce three positive results in 4 tests, while another might require 9 tests. The fixed-number-of-trials condition therefore fails.
The count recorded at stopping is not a binomial count. The fact that the individual test outcomes may have a constant probability and be independent does not repair the failure of the fixed-\(n\) condition. If staff instead tested exactly 20 people and counted the positive results among those 20, then the number of trials would be fixed; the remaining conditions would still need to be checked.
Common Mistakes and a Practical Checklist
- Assuming that every count is binomial. A count of successes qualifies only when all four conditions are reasonable.
- Calling a sample without replacement automatically independent. When the population contains a mix of outcomes, such draws generally are dependent. Use the 10% guideline only to support an approximation when its setting applies.
- Treating a past rate as proof of a constant probability. An observed fraction can provide a working estimate, but stability over the trials must come from knowledge of the process.
- Ignoring the stopping rule. Stopping after a specified number of successes makes the number of trials variable; it is not the same as fixing \(n\) in advance.
- Defining success by whether an outcome is desirable. Success is simply the outcome being counted. For a defect count, a defective item can be coded as success.
Before using a binomial model, state the trial, define success, identify the fixed value of \(n\), and explain why one value of \(p\) and independence are plausible. If sampling without replacement from a finite population, check the sample fraction and describe the model as approximate when using the 10% guideline. If a condition is seriously violated, do not present a binomial calculation as exact.
Check Your Understanding
For each situation, identify which binomial conditions hold and which need further justification.
- A technician tests exactly 30 devices and counts how many pass. What details are needed to assess whether the count is binomial?
- A researcher samples 40 people without replacement from a population of 2,000. What does the 10% guideline say, and does it make the draws exactly independent?
- A clinic enrolls patients until it has observed 15 who meet a specified criterion. Which binomial condition fails for the count collected at that stopping time?
- A factory has a historical defect fraction of 0.02. Why does this alone not establish a constant success probability for the next 100 items?
- Why can sampling without replacement from a population in which everyone has the same yes/no outcome still produce independent trial outcomes?