A Checklist Turns Recognition into a Justification
In Recognizing a Binomial Setting, you met the four features a count must have to follow a binomial model. The BINS checklist gives those features a compact order: Binary outcomes, Independent trials, a fixed Number of trials, and the Same probability of success on every trial.
A checklist is useful because it prompts you to justify each condition from the situation rather than deciding from one appealing detail. A process might have two outcomes on every trial and a fixed trial count, yet still fail because the trials affect one another or their success probabilities change. A clear verdict connects each condition to the details provided.
Before starting the checklist, define one trial, identify success, and say what \(X\) counts. This prevents a common mix-up: the number of trials \(n\) is set by the process, while the number of successes \(X\) is the result that varies. If all four conditions are met, describe the model as \(X\sim\operatorname{Binomial}(n,p)\), where \(p\) is the common probability of success on each trial.
Apply BINS in Order
For each condition, look for evidence in the description. Sometimes the prompt states a condition directly, such as “the outcomes are independent.” In other situations, you must explain why it seems reasonable or why a feature of the process makes it fail. When the information is insufficient, say so rather than claiming the condition is established.
Can each trial be classified in exactly two relevant ways, success or failure? State what success means for this question. Other details may exist, but the trial must have just two categories relevant to the count.
Does the outcome of one trial leave the probabilities for other trials unchanged? Look for shared conditions, outcomes that affect later trials, or sampling without replacement from a small group.
Is the number of trials \(n\) decided before the process begins? A fixed number of successes is not the same as a fixed number of trials: a process that continues until a target is reached has a variable trial count.
Does every trial have the same probability \(p\) of success? Check whether the people, conditions, or selection process change across trials. A fixed trial count alone does not guarantee a common success probability.
Independence and the same probability are related, but they ask different questions. Independence asks whether trials affect one another. The same-probability condition asks whether each trial has the same chance of success. A setting can satisfy one and fail the other, so give them separate explanations.
In practice, use three possible verdicts for an individual condition: met, not met, or reasonable as an approximation. For example, selecting without replacement makes trials technically dependent, but a binomial approximation may be reasonable when the sample is small relative to the population. When sampling without replacement, the AP Statistics 10% condition is a common guideline: the sample size should be no more than 10% of the population.
Worked Example: Monitoring a Sensor
Worked Example: Monitoring a Sensor
A building manager records whether a particular air-quality sensor gives a warning on each of 18 scheduled, separate days. For a model, assume the sensor’s daily warning chance is 0.08 and one day’s result does not affect another’s. Let \(X\) be the number of days with a warning. Is a binomial model appropriate?
State. One trial is one scheduled day. Define success as the sensor giving a warning. The variable \(X\) counts warning days among the 18 days.
Plan. Apply BINS to the daily results: check for two outcomes, independence, a fixed number of days, and the same warning probability.
Do. Binary: Each day is classified as warning or no warning. Independent: The model explicitly assumes that one day’s result does not affect another’s. Number fixed: The manager records exactly 18 scheduled days. Same probability: The model specifies a warning probability of 0.08 on every day.
Conclude. All four BINS conditions are satisfied under the stated assumptions, so a binomial model is appropriate: \(X\sim\operatorname{Binomial}(18,0.08)\). Here, \(n=18\) is the number of days and \(p=0.08\) is the chance of a warning on each day. This verdict depends on the model assumptions, including that the sensor’s daily chance does not change.
Notice how the justification identifies evidence for each letter. Saying only “there are 18 days, so it is binomial” would not be enough: the number of days supports the fixed-\(n\) condition but says nothing by itself about independence or the warning probability.
Worked Example: Selecting Students Without Replacement
Worked Example: Selecting Students Without Replacement
A class has 30 students, of whom 12 bike to school. A teacher randomly selects 6 different students, without replacement, and counts how many bike to school. Let \(Y\) be that count. Is a binomial model appropriate?
State. One trial is selecting one student. Define success as selecting a student who bikes to school. The variable \(Y\) counts students who bike among the 6 selected.
Plan. Check all four BINS conditions. Pay particular attention to whether selecting a student changes the chances on later selections and whether the sample is small enough for the 10% guideline.
Do. Binary: Each selected student either bikes or does not. Independent: The selections are not independent because they are made without replacement. For example, after selecting a student who bikes, fewer biking students remain among those available. Also, 6 is 20% of the class, which exceeds the 10% guideline: \(6/30=0.20\). Independence is not reasonable to treat as approximately satisfied by that guideline. Number fixed: Exactly 6 students are selected. Same probability: The probability changes as students are removed. It begins at \(12/30\); after a selection, the numbers of biking students and students remaining depend on who was selected.
Conclude. A binomial model is not appropriate for \(Y\) as described because the independence and same-probability conditions fail. There are two outcomes per selection and a fixed sample size, but those features alone do not make the count binomial.
The 10% guideline does not turn sampling without replacement into literal independence. It is used to judge whether the dependence may be small enough for a binomial approximation. Here, the sample is a substantial fraction of the class, so the changing selection probabilities matter.
Worked Example: A Small Sample from a Large Lot
Worked Example: A Small Sample from a Large Lot
A warehouse contains 800 packages, and about 4% of the packages in the lot are incorrectly sealed. An inspector randomly selects 40 packages without replacement and counts the incorrectly sealed ones. Let \(Z\) be that count. Would a binomial model be reasonable?
State. One trial is inspecting one selected package. Define success as finding an incorrectly sealed package. The variable \(Z\) counts incorrectly sealed packages among the 40 inspected.
Plan. Check BINS, distinguishing exact independence from a reasonable approximation. For sampling without replacement, compare the sample size with 10% of the lot.
Do. Binary: A package is incorrectly sealed or it is not. Independent: Because selection is without replacement, the trials are not exactly independent. However, 10% of 800 is 80, and the sample of 40 is no more than 80. Thus, the sample is at most 10% of the lot, so treating the trials as approximately independent is reasonable by the 10% guideline. Number fixed: The inspector selects exactly 40 packages. Same probability: The chance changes slightly as packages are removed, but the small sample relative to the lot makes using the approximate common probability \(p=0.04\) reasonable.
Conclude. A binomial model is reasonable as an approximation: \(Z\) can be modeled approximately by \(\operatorname{Binomial}(40,0.04)\). This is not an exact claim of independence or an exactly constant probability; it is a judgment supported by the sample being no more than 10% of the lot and by the stated approximate defect rate.
This example shows why a good BINS response can qualify its verdict. “The trials are independent” would overstate the situation. A more accurate statement is that selection without replacement creates dependence, but the 10% condition supports treating that dependence as small enough for an approximate binomial model.
Worked Example: Independent Trials with Changing Chances
Worked Example: Independent Trials with Changing Chances
A packaging machine seals 12 packages in sequence. Assume each package’s seal result is independent of the others. Due to a planned machine adjustment, the probability of a correct seal is 0.96 for each of the first 6 packages and 0.99 for each of the last 6. Let \(W\) be the number of correctly sealed packages. Is \(W\) binomial?
State. One trial is sealing one package. Define success as a correct seal. The variable \(W\) counts correctly sealed packages among the 12 packages.
Plan. Check each BINS condition separately, especially whether all 12 trials share the same probability of success.
Do. Binary: Each package is correctly sealed or incorrectly sealed. Independent: Independence is stated in the situation. Number fixed: The machine seals exactly 12 packages. Same probability: This condition fails. The chance of a correct seal is 0.96 on the first 6 trials and 0.99 on the last 6, so there is no single common value of \(p\) for all 12 trials.
Conclude. \(W\) is not binomial as described because the same-probability condition fails. Independence, binary outcomes, and a fixed number of trials do not compensate for changing success probabilities.
This is a useful reminder that a change in conditions can matter even when every trial still has the same two outcomes. If a problem instead said the adjustment had no effect on success chances, that would be a different model assumption to evaluate.
Common Mistakes and Full-Credit Communication
For a strong response, define the count and state what the evidence says about each relevant condition. If a condition fails, name it and point to the detail causing the failure. If a condition is only plausible as an approximation, say that rather than presenting the model as exact.
- Using “binary” as the whole argument. A yes-or-no outcome does not establish independence, a fixed trial count, or the same probability. Apply all four letters.
- Confusing a fixed target with a fixed trial count. “Continue until 4 packages are found” fixes the number of successes, not the number of packages inspected. The trial count can vary.
- Combining independence and same probability. In the machine example, the outcomes are independent, but the probability changes. Name these as separate conditions.
- Ignoring sampling without replacement. Removing selected people or objects can change the chances on later selections. Consider the population size and the 10% condition before claiming independence is reasonable.
- Giving parameters without defining them. When a binomial model is appropriate, say what \(n\) counts and what event \(p\) represents. For the sensor, \(n=18\) days and \(p=0.08\) chance of a warning per day.
- Claiming more than the prompt supports. If independence or a constant probability is an assumption, call it an assumption. If the information is missing, explain what would need to be known rather than silently treating the condition as true.
A concise full-credit explanation for the sensor setting would identify the day as a trial, define a warning as success, and state that there are two outcomes, 18 fixed days, independent results by assumption, and a common warning probability of 0.08. For the class sample, it would identify the two outcomes and fixed sample size, then explain that sampling without replacement from 30 students changes the chances and that the sample exceeds 10% of the class.
Key Takeaway
Use BINS to give a separate verdict for Binary outcomes, Independence, a fixed Number of trials, and the Same success probability. A single failed condition rules out a binomial model as described; an approximation should be identified as an approximation and supported by the situation.
Check Your Understanding
For each setting, define the count and use BINS to explain whether a binomial model is appropriate, exact, or reasonable only as an approximation.
- A gardener checks 15 seedlings. Each is classified as alive or not alive, and the checks are independent with the same 0.82 chance of being alive. Define the count and state its binomial parameters.
- A volunteer asks people to donate until 10 people have agreed. Which BINS condition is not met for the number of people asked?
- A club has 40 members, including 16 who use a bicycle to get to meetings. A coordinator randomly selects 4 members without replacement. Explain whether the 10% guideline supports treating the selections as approximately independent.
- A factory tests 25 parts independently. The chance a part passes is 0.98 for the first 10 parts and 0.95 for the remaining 15. Which BINS conditions hold, and which fails?
- A student says, “There are 20 trials and two outcomes each, so the count is binomial.” Identify the other two conditions the student must check and describe what each asks.