What changes when you learn an event occurred?
You will be able to: Calculate a conditional probability using the restricted reference group.
What changes when you learn an event occurred?
In a club of 40, 18 students prefer digital notes. If you learn the selected student attends the morning session, you should look only at that session’s 20 members.
A useful starting point: Can two events happen in the same trial? →
Words and symbols before equations
- P(A given B)
- Probability of A after restricting to B, often written P(A∣B).
- Given event
- The information defining the denominator.
- Joint event
- Both A and B occur.
- Conditional sample space
- The outcomes still eligible after the information is known.
What this picture assumes
P(A)=0.40 and P(B)=0.30 are fixed. The intersection ranges from 0 to 0.30. Rectangular region areas encode probabilities in this synthetic model; it is not a measured population.
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.
- Union 0.58; intersection 0.12. Events are not disjoint and independent in this model. Independence needs intersection 0.12, not zero.
- 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 morning group includes 12 digital and 8 paper preferences, so P(digital given morning)=12/20=0.60. The unconditional digital probability is 18/40=0.45.
The general formula is P(A∣B)=P(A∩B)/P(B), provided P(B)>0. Dividing removes the probability mass outside the given event and rescales B to a whole.
The order matters. P(morning given digital)=12/18=2/3 instead of 0.60. Also, a condition describes information; it need not be an event occurring earlier in time.
A worked example, step by step
Suppose P(A∩B)=0.12 and P(B)=0.30. Find P(A given B).
- The requested condition is B, so B supplies the denominator.
- Verify P(B)=0.30 is positive.
- Divide 0.12/0.30=0.40.
- Among outcomes in B, 40% also belong to A; the joint probability remains 12% of all outcomes.
The probability of “A and B” uses the whole sample space; “A given B” uses only B.
If 6 of 15 B members are also A, what is P(A given B)?
Compare with an explanation
6/15=0.40, using the B group as the reference set.
Predict. Change one thing. Explain.
Vary the overlap while keeping P(A)=0.40 and P(B)=0.30 fixed. Predict P(A given B) and explain why P(B given A) generally differs.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Union 0.58; intersection 0.12. Events are not disjoint and independent in this model. Independence needs intersection 0.12, not zero.
| Quantity | Calculation | Value |
|---|---|---|
| P(A or B) | 0.40 + 0.30 − 0.12 | 0.58 |
| P(A given B) | 0.12 / 0.30 | 0.4 |
| P(B given A) | 0.12 / 0.40 | 0.3 |
| Independence target | 0.40 × 0.30 | 0.12 |
P(A)=0.40 and P(B)=0.30 are fixed. The intersection ranges from 0 to 0.30. Rectangular region areas encode probabilities in this synthetic model; it is not a measured population.
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 table of 50 students contains 20 cyclists, including 8 who prefer morning classes. There are 25 morning-preferring students overall. Find both directions of conditioning.
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Compare with the answer and four-point rubric
- 1 point: P(morning given cyclist)=8/20=0.40.
- 1 point: P(cyclist given morning)=8/25=0.32.
- 1 point: The joint share is 8/50=0.16.
- 1 point: Different reference groups explain why the two conditional probabilities differ.
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 “given” do?
Restricts the reference group.
RECALL 2Must the condition happen earlier in time?
No; it specifies information.
RECALL 3When is the elementary conditional ratio undefined?
When the conditioning event has probability zero.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What changes when you learn an event occurred?
- P(A∣B)=P(A∩B)/P(B), P(B)>0.
- Reverse conditioning usually changes the denominator.
- A zero-probability condition cannot use this ratio.
Remember: The probability of “A and B” uses the whole sample space; “A given B” uses only B.
Conditions: P(A)=0.40 and P(B)=0.30 are fixed. The intersection ranges from 0 to 0.30. Rectangular region areas encode probabilities in this synthetic model; it is not a measured population.
Refresh Kid · AP Statistics Unit 2 · Objectives 2.6.A · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.6, objectives 2.6.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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