What should repeated success counts look like?
You will be able to: Interpret binomial mean and standard deviation and compare a simulation to the exact model.
What should repeated success counts look like?
Across many sets of 20 independent attempts with p=0.30, the success count averages 6. Individual sets still vary.
A useful starting point: How do you count all the ways to get k successes? →
Words and symbols before equations
- Binomial mean μ=np
- Long-run average success count per n-trial set.
- Binomial variance np(1−p)
- The squared spread of the count distribution.
- Binomial standard deviation
- √[np(1−p)], in successes per set.
- Simulation replication
- One full n-trial set producing one count.
What this picture assumes
Each repetition generates n independent Bernoulli trials with a common p. Teal bars show exact probabilities; orange outlines show simulated relative frequencies. More repetitions do not guarantee closer agreement at every step.
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 = 2 successes; SD √[np(1−p)] = 1 successes. 200 simulated repetitions; empirical mean 1.845.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Each trial contributes an expected p successes, so n trials contribute np. Independence allows the trial variances p(1−p) to add, giving variance np(1−p).
For n=20,p=0.30, μ=6 and σ=√4.2≈2.049 successes per set. Neither says a set must contain 6 successes or lie within 2.049 of 6.
In a simulation, generate n Bernoulli outcomes per replication, count successes, and repeat many times. Compare relative frequencies to theoretical binomial bars. Increasing replications refines the estimate; changing n changes the underlying distribution.
A worked example, step by step
For a binomial X with n=16,p=0.25, calculate the mean and standard deviation and interpret 40 simulated sets with X≥6 out of 200.
- μ=16×0.25=4 successes per set.
- Variance=16×0.25×0.75=3.
- σ=√3≈1.732 successes per set.
- The simulation estimates P(X≥6) as 40/200=0.20; this finite estimate need not equal the exact binomial tail.
The number of trials n controls each count. The number of simulated sets controls approximation precision.
Is the mean necessarily an integer count?
Compare with an explanation
No. For example n=3,p=0.5 gives mean 1.5, although individual counts are integers.
Predict. Change one thing. Explain.
Change the number of simulated sets and the seed. Compare the empirical distribution with the exact binomial probabilities and explain finite variation.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Mean np = 2 successes; SD √[np(1−p)] = 1 successes. 200 simulated repetitions; empirical mean 1.845.
| k successes | Exact probability | Simulated frequency |
|---|---|---|
| 0 | 0.0625 | 0.09 |
| 1 | 0.25 | 0.275 |
| 2 | 0.375 | 0.375 |
| 3 | 0.25 | 0.22 |
| 4 | 0.0625 | 0.04 |
Each repetition generates n independent Bernoulli trials with a common p. Teal bars show exact probabilities; orange outlines show simulated relative frequencies. More repetitions do not guarantee closer agreement at every step.
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 count has n=25,p=0.4. Find μ, variance and σ. Interpret 30 qualifying counts among 100 simulations.
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Compare with the answer and four-point rubric
- 1 point: μ=10.
- 1 point: Variance=25×0.4×0.6=6.
- 1 point: σ=√6≈2.449 successes per set.
- 1 point: The qualifying-event estimate is 30/100=0.30 under the simulation assumptions; it is not a guaranteed next-set result.
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 unit does binomial σ have?
Number of successes per fixed-size set.
RECALL 2What happens at p=0 or p=1?
The count is constant and σ=0.
RECALL 3Does more simulation change the theoretical mean?
No; it changes the precision of the empirical approximation.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What should repeated success counts look like?
- μ=np.
- σ=√[np(1−p)].
- Simulated probability = qualifying sets / all sets.
Remember: The number of trials n controls each count. The number of simulated sets controls approximation precision.
Conditions: Each repetition generates n independent Bernoulli trials with a common p. Teal bars show exact probabilities; orange outlines show simulated relative frequencies. More repetitions do not guarantee closer agreement at every step.
Refresh Kid · AP Statistics Unit 2 · Objectives 2.10.B, 2.10.C, 2.10.D · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.10, objectives 2.10.B, 2.10.C, 2.10.D. 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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