How does a chance outcome become a number?
You will be able to: Define a discrete random variable and construct its probability distribution.
How does a chance outcome become a number?
Toss a fair coin twice and count the heads. The sequences HH, HT, TH and TT are outcomes, while the head count X is a number assigned to each outcome.
A useful starting point: How do you avoid counting the overlap twice? →
Words and symbols before equations
- Random variable X
- A numerical function of a random outcome.
- Discrete
- Possible values form a finite or countable set.
- Probability distribution
- Every possible value paired with its probability.
- P(X=x)
- Probability that the variable takes a particular value x.
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 mapping is HH→2, HT→1, TH→1 and TT→0. Different outcomes can yield the same random-variable value, so combine their probabilities.
The distribution is P(X=0)=1/4, P(X=1)=1/2 and P(X=2)=1/4. Each probability is nonnegative and the complete set sums to 1.
A bar chart shows probability at each discrete value. Bar height is probability here. This differs from a continuous density curve, where interval area—not height—is probability.
A worked example, step by step
Three equally likely cards carry 0,0,3 bonus points. Let X be the points on a uniformly selected card. Construct its distribution.
- There are three equally likely physical cards, even though two labels match.
- Possible X values are 0 and 3.
- P(X=0)=2/3 and P(X=3)=1/3.
- The probabilities sum to 1; do not assign 1/2 to each distinct label.
Equally likely elementary outcomes do not make the distinct random-variable values equally likely.
Can X count successes while the elementary outcomes are sequences?
Compare with an explanation
Yes. X maps each sequence to its success count.
Predict. Change one thing. Explain.
Inspect a three-value bonus distribution and change its high-bonus probability. Verify that the probability table still sums to 1 and distinguish outcomes from numerical values.
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 questionA spinner has four equal sectors labeled 0,1,1,4. Define X as the displayed number and give its distribution.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: X has possible values 0,1,4.
- 1 point: P(X=0)=1/4.
- 1 point: P(X=1)=2/4 and P(X=4)=1/4.
- 1 point: The three probabilities sum to 1, with value 1 more likely because it labels two sectors.
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 a random variable do?
Maps each outcome to a number.
RECALL 2What checks validate a probability distribution?
All probabilities are nonnegative and the complete total is 1.
RECALL 3Why can distinct values have unequal probabilities?
Different numbers or weights of outcomes can produce them.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How does a chance outcome become a number?
- List every possible x and P(X=x).
- Probabilities are nonnegative and sum to 1.
- Combine probabilities of outcomes mapping to the same value.
Remember: Equally likely elementary outcomes do not make the distinct random-variable values equally likely.
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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