How can repeated trials estimate a probability?
You will be able to: Design and interpret a simulation with a clearly defined event and trial.
How can repeated trials estimate a probability?
A digital badge appears with probability 0.30 on each independent attempt. You want the chance of at least one badge in three attempts.
A useful starting point: Which total belongs in the denominator? →
Words and symbols before equations
- Trial
- One complete repetition of the chance process being studied.
- Simulation
- A chance model that imitates specified assumptions.
- Relative frequency
- Number of event occurrences divided by the number of trials.
- Seed
- A value that makes a pseudo-random demonstration reproducible.
What this picture assumes
Each trial has three independent attempts, each with success probability 0.30. A trial counts as an event if at least one succeeds. A seeded pseudorandom simulation is an illustration, not real data.
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.
- 122 events in 200 trials: estimate 0.61. Exact probability 1−0.7³ = 0.657 (orange horizontal line). Current absolute error 0.047.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Use equally likely digits 0–9, letting 0,1,2 mean badge and 3–9 mean no badge. One trial consists of three independent digits, not one digit. Record whether at least one badge appears.
Repeat the full three-attempt trial many times. If the event occurs in 64 of 100 trials, estimate its probability as 0.64. The exact model value is 1−0.7³=0.657, useful for checking the simulation.
The law of large numbers concerns long-run relative frequency under stable independent trials. Estimates need not move closer at every step. More trials improve simulation precision; they do not repair a wrong probability or a false independence assumption.
A worked example, step by step
In 200 simulated three-attempt trials, 129 contain at least one badge. Give the estimate and describe the model.
- Assign digits 0–2 to badge and 3–9 to no badge.
- Generate three independent digits per trial and check for at least one success.
- Repeat 200 trials and divide 129 by 200 to obtain 0.645.
- The estimate is 64.5% under the chosen model; a different seed can change the finite result.
The number of simulated trials is different from the number of attempts within each trial.
After five failures, is the next independent attempt more likely to succeed?
Compare with an explanation
No. Under the model its success probability remains 0.30.
Predict. Change one thing. Explain.
Increase the number of trials while holding the seed fixed, then change the seed. Compare the running estimate with 0.657 and find a segment where error increases temporarily.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
122 events in 200 trials: estimate 0.61. Exact probability 1−0.7³ = 0.657 (orange horizontal line). Current absolute error 0.047.
| Method | Probability |
|---|---|
| Exact complementary calculation | 0.657 |
| Simulation estimate | 0.61 |
Each trial has three independent attempts, each with success probability 0.30. A trial counts as an event if at least one succeeds. A seeded pseudorandom simulation is an illustration, not real data.
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 questionDesign a digit simulation for at least one success in two independent attempts with p=0.20. Interpret 70 event trials out of 200.
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Compare with the answer and four-point rubric
- 1 point: Use digits 0,1 as success and 2–9 as failure.
- 1 point: Generate two independent digits for one trial; record whether either is a success.
- 1 point: Repeat 200 times; estimate 70/200=0.35.
- 1 point: Exact model probability is 1−0.8²=0.36; the finite simulation need not equal it.
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 defines one simulation trial?
One full repetition of the process whose event is being estimated.
RECALL 2Does the estimate improve at every added trial?
No; it fluctuates.
RECALL 3What does a fixed seed provide?
A reproducible pseudo-random demonstration, not certainty about reality.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How can repeated trials estimate a probability?
- Estimate = event trials / all trials.
- State outcome mapping, one-trial rule, event and repetition count.
- Long-run stability does not mean short-run compensation.
Remember: The number of simulated trials is different from the number of attempts within each trial.
Conditions: Each trial has three independent attempts, each with success probability 0.30. A trial counts as an event if at least one succeeds. A seeded pseudorandom simulation is an illustration, not real data.
Refresh Kid · AP Statistics Unit 2 · Objectives 2.3.A · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.3, objectives 2.3.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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