When is a count binomial?
You will be able to: Check every condition before using a binomial model.
When is a count binomial?
A player takes 5 shots, each with a modeled success probability of 0.60. Counting made shots can be binomial if the assumptions hold.
A useful starting point: How variable is a chance outcome? →
Words and symbols before equations
- Success
- The outcome being counted, not necessarily a desirable result.
- Binomial random variable
- Number of successes in a fixed number of independent binary trials with common p.
- n
- The fixed number of trials.
- p
- The probability of success on every trial.
What this picture assumes
A fixed number n of independent, identical trials, each with success probability p, is assumed. X counts successes. A cutoff beyond n is allowed: its probability follows the selected event, not a truncated count.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Mean np = 1.2 successes; SD √[np(1−p)] = 0.849 successes. P(X = 2) = 0.288.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Check a fixed trial count, two outcome categories per trial, independence and a constant success probability. Define what counts as success before calculating.
Real shots may violate the model if fatigue changes p or previous results affect later performance. The binomial formula is exact only for its assumptions, not for every story involving counts.
Sampling without replacement makes outcomes dependent. When a random sample is at most about 10% of a finite population, a binomial approximation is often reasonable for success counts because depletion is limited; identify it as approximate. Stopping at the first success is not a fixed-n binomial process.
A worked example, step by step
Choose 5 items without replacement from a box of 8 containing 3 defective items. Is the defective count exactly binomial?
- The trial count is fixed at 5 and each item is defective or not.
- But removal changes the remaining defective proportion.
- The draws are not independent, and 5 is far more than 10% of 8.
- Therefore an exact binomial model is inappropriate; use a without-replacement model or direct conditional reasoning.
A binary outcome alone is insufficient. Independence, fixed n and common p all matter.
Is counting attempts until the first success binomial?
Compare with an explanation
No. The number of attempts is not fixed in advance.
Predict. Change one thing. Explain.
Change n and p in an explicitly independent model. Identify which assumptions the controls hold fixed and which real-world assumptions the display cannot verify.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Mean np = 1.2 successes; SD √[np(1−p)] = 0.849 successes. P(X = 2) = 0.288.
| k successes | P(X=k) | Included in request? |
|---|---|---|
| 0 | 0.216 | No |
| 1 | 0.432 | No |
| 2 | 0.288 | Yes |
| 3 | 0.064 | No |
A fixed number n of independent, identical trials, each with success probability p, is assumed. X counts successes. A cutoff beyond n is allowed: its probability follows the selected event, not a truncated count.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the probability values, reference groups, graph scales or model assumptions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA machine independently labels each of 12 packages incorrectly with probability 0.02. Define a binomial X and state all conditions.
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Compare with the answer and four-point rubric
- 1 point: X counts incorrectly labeled packages among the 12.
- 1 point: n=12 is fixed; each package is incorrect or correct.
- 1 point: The problem states independent labeling and common p=0.02.
- 1 point: Thus X has a binomial model with n=12 and p=0.02, conditional on those assumptions.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does n mean?
Fixed number of trials.
RECALL 2What does p mean?
Common success probability per trial.
RECALL 3Why can without-replacement draws violate binomial assumptions?
Removing items changes composition and creates dependence.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
When is a count binomial?
- Binomial: fixed n, binary trials, independent trials, common p.
- X counts successes, from 0 through n.
- State when using an approximation for a finite population.
Remember: A binary outcome alone is insufficient. Independence, fixed n and common p all matter.
Conditions: A fixed number n of independent, identical trials, each with success probability p, is assumed. X counts successes. A cutoff beyond n is allowed: its probability follows the selected event, not a truncated count.
Refresh Kid · AP Statistics Unit 2 · Objectives 2.10.A · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.10, objectives 2.10.A. Framework effective Fall 2026, checked September 17, 2026. Unit 2 includes probability, random variables, probability models and introductory sampling distributions; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Assumptions about independence, replacement and equal likelihood are stated before calculations. Simulation estimates fluctuate. Discrete probability is summed; continuous probability is area. Sampling distributions and randomization distributions use different repetition mechanisms. Formal inference comes in later units.
The Organic Chemistry Tutor companion title and destination were checked; the full video was not reviewed. Khan Academy’s destination was checked, but its lesson content was not fully readable by the research tool. OpenStax provides optional reference reading. No provider scripts, questions or graphics were copied. Refresh Kid is not affiliated with these providers.
GitHub’s 3D website collection informed optional spatial inspection. Our original three-toss outcome cube uses self-hosted Three.js with its MIT license. Eight corners represent eight equally likely sequences of three fair independent tosses. Conditioning removes ineligible sequences; camera rotation never changes probabilities. Quantitative graphs remain 2D to avoid perspective distortion. Complete labeled diagrams, outcome lists and explanations remain available without 3D.
Independent teacher review and observation of students remain pending. Technical checks do not certify statistical accuracy, accessibility or learning effectiveness. This is a review edition.
Released AP Statistics questions and scoring guides are optional. Older exams use the earlier framework, so check alignment before selecting parts. All practice on this page is original, not official AP material.
Learn → Explore → Practice → Review is informed by the IES learning guide. This implementation has not yet been evaluated with learners.
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