Algebra, conditions and vectors
Book Isle conditional probability: changing the given condition changes the denominator
Use twelve original cards to distinguish an intersection, both conditional directions and joint probability under an equal-likelihood assumption.
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“Stars among red cards” and “red among star cards” share an intersection but can have different denominators. Read an original card exercise in Book Isle and record restricted sample spaces in available notes or externally. Readers calculate; the app is not assumed to provide a probability solver, and model counts are not population statistics.
1. Give twelve cards unique identities
List twelve equally likely cards. Red cards are 1–6 and star cards 1,2,3,7; membership can overlap. Retain identities before calculating because marginal totals alone do not determine the intersection. This is an original model, not empirical sampling or a multi-draw replacement problem.
All: 1,2,3,4,5,6,7,8,9,10,11,12
Red R: 1,2,3,4,5,6
Star S: 1,2,3,7
R AND S: 1,2,3
2. Count the intersection and four regions
The four disjoint regions contain three both, three red-only, one star-only and five neither. Their total is twelve, recovering red six and star four. Place every numbered card in exactly one region so the intersection is not double-counted or duplicated across non-overlapping portions. Within red cards, no-star cards 4,5,6 give 3/6; this complements 3/6 to one under the same condition. The 1/4 of non-red among star cards is a different restricted space. List both complementary groups under the same condition before adding fractions.
3. Condition on red cards
Given red, count only six red cards: three have stars, so P(S|R)=3/6=1/2. OpenStax’s conditional-probability definition requires a positive conditioning probability. Read the bar as “given,” state the restricted group before choosing six, and do not divide by twelve while calling it conditional.
4. Reverse the given condition and recount
Given star, restrict to four cards, three red: P(R|S)=3/4. The joint probability is 3/12=1/4. Label each question and denominator set separately. A shared numerator does not make the conditional directions or the joint probability equal, and the conditional bar is not freely interchangeable.
P(S|R)=3/6=1/2
P(R|S)=3/4
P(R AND S)=3/12=1/4
5. Verify the formula using original fractions
Verify (3/12)/(6/12)=1/2 and (3/12)/(4/12)=3/4. Since P(S)=1/3 differs from P(S|R), independence cannot be assumed in this model. This uses complete stipulated counts, not one observed draw. An empty conditioning event violates the positive-denominator condition; do not manufacture zero by dividing by zero.
6. Save question wording and a reading anchor
Open the owned page through available reading controls and retain given condition, requested event and denominator together. Use external notes if tables/formulas are unsuitable; no automatic Venn tool is assumed. Save a chapter/page and short anchor, reopen and read the question before its answer. Wrapping changes must not change card identities or counts.
7. Troubleshoot scope, double counts and independence
If both conditional results are 1/4, check an unchanged denominator of twelve. A total of fifteen suggests double-counting overlap. Different category names do not imply independence; compare conditional and original proportions. Retain error/correction and never delete a card merely to force a result. A changed model needs a new version and complete recount.
8. Reopen and deliver three reproducible questions
Reopen with fractions hidden and recompute all three from identities, checking total twelve. Deliver equal-likelihood assumptions, labels, intersection, restricted groups and formula checks. Explain denominators six/four/twelve. Unequal selection weights require a different probability model; unperformed app/device steps remain pending and stipulated counts are not empirical validation.
Related tutorials and sources
Exclusive versus independent events, Function composition.
Checked on November 14, 2026 against the official Book Isle App Store listing and Apple public lookup: Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. Use existing reading, available notes or external records and verify controls in your installed version. Original samples and manual calculations are not new automatic analysis features. No physical-device, private-file or universal export-environment testing is claimed.