How do “less than,” “at most” and “at least” differ?
You will be able to: Build a cumulative table and translate inequalities for a discrete variable.
How do “less than,” “at most” and “at least” differ?
A bonus X is 0, 2 or 5 points with probabilities 0.50, 0.30 and 0.20. “At most 2” includes both 0 and 2, while “less than 2” includes only 0.
A useful starting point: How does a chance outcome become a number? →
Words and symbols before equations
- Cumulative distribution F(x)
- P(X≤x), the probability at or below x.
- At most k
- X≤k.
- At least k
- X≥k.
- Strict inequality
- Excludes the boundary value.
What this picture assumes
X is a hypothetical reward in points taking values 0, 2 or 5. P(X=2)=0.30 is fixed; P(X=0)=0.70−P(X=5). Probabilities remain nonnegative and sum to 1.
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 1.6 points; SD 1.908 points. P(X ≤ 2) = 0.8. The mean is a long-run average and need not be a possible single reward.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
The cumulative totals are F(0)=0.50, F(2)=0.80 and F(5)=1. Between possible values the cumulative probability stays constant, producing a step function.
For a discrete variable, including a boundary with positive probability changes the answer. P(X<2)=0.50, while P(X≤2)=0.80.
For integer-valued X, P(X≥k)=1−P(X≤k−1). The complement of “at least 2” is “less than 2,” not “at most 2.” List included values when uncertain.
A worked example, step by step
For X=0,2,5 with probabilities 0.5,0.3,0.2, find P(X≥2), P(X>2) and F(3).
- At least 2 includes values 2 and 5, so probability 0.3+0.2=0.5.
- More than 2 includes only 5, so probability 0.2.
- At or below 3 includes 0 and 2, giving F(3)=0.8.
- The gap between 2 and 5 contains no possible X values, so F is flat there.
For discrete values, boundary inclusion matters. Do not use 1−F(k) for P(X≥k).
What is F(4) in the example distribution?
Compare with an explanation
0.80, because only values 0 and 2 are at or below 4.
Predict. Change one thing. Explain.
Move the cutoff across values that are possible and values in between. Compare the at-or-below shaded bars and cumulative table.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Mean 1.6 points; SD 1.908 points. P(X ≤ 2) = 0.8. The mean is a long-run average and need not be a possible single reward.
| x (points) | P(X=x) | F(x)=P(X≤x) | x × P(X=x) | (x−μ)² × P(X=x) |
|---|---|---|---|---|
| 0 | 0.5 | 0.5 | 0 | 1.28 |
| 2 | 0.3 | 0.8 | 0.6 | 0.048 |
| 5 | 0.2 | 1 | 1 | 2.312 |
| Summary | Value |
|---|---|
| Mean μ (points) | 1.6 |
| Variance (points²) | 3.64 |
| Standard deviation (points) | 1.908 |
| P(X ≤ 2) | 0.8 |
X is a hypothetical reward in points taking values 0, 2 or 5. P(X=2)=0.30 is fixed; P(X=0)=0.70−P(X=5). Probabilities remain nonnegative and sum to 1.
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 questionX takes 0,1,2 with probabilities 0.2,0.5,0.3. Give F(0), F(1), P(X≥1) and P(X>1).
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Compare with the answer and four-point rubric
- 1 point: F(0)=0.2.
- 1 point: F(1)=0.2+0.5=0.7.
- 1 point: P(X≥1)=0.8.
- 1 point: P(X>1)=0.3; the difference is the mass at X=1.
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 F(x) include?
The boundary x and all smaller values.
RECALL 2Can F decrease?
No.
RECALL 3Why is a discrete CDF flat between values?
No new probability mass is encountered.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do “less than,” “at most” and “at least” differ?
- F(x)=P(X≤x).
- P(X>k)=1−F(k).
- For integer X, P(X≥k)=1−F(k−1).
Remember: For discrete values, boundary inclusion matters. Do not use 1−F(k) for P(X≥k).
Conditions: X is a hypothetical reward in points taking values 0, 2 or 5. P(X=2)=0.30 is fixed; P(X=0)=0.70−P(X=5). Probabilities remain nonnegative and sum to 1.
Refresh Kid · AP Statistics Unit 2 · Objectives 2.8.A · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.8, objectives 2.8.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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