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UAT-UK SPEC: M7MODULE: MATHEMATICS 1TOPIC ID: ARCH-M1-12

Multi-Stage Conditional Probability Trees (Without Replacement) & Algebraic Venn Overlaps

Master Multi-Stage Conditional Probability Trees (Without Replacement) & Algebraic Venn Overlaps on the ESAT Mathematics 1 paper. Traditional classroom derivations waste valuable seconds. Using the 2-step Sub-90 invariant elimination method compresses solve time to 40s, saving +100s per question.

Standard Classroom Time
140s
Textbook multi-line derivations bleed exam clock
Sub-90 Target Pace
40s
Comfortably below the 88.8s per-question limit
Time Saved for Review
+100s
Banks clock buffer across 27 questions

2-Step Sub-90 Shortcut Formula

How to eliminate intermediate algebra and solve under 45 seconds.

Launch Interactive Lesson
STEP 1 • SETUP INVARIANT (LINE 1)10–15s
Line 1 (Unordered pairs shortcut): 5×4+3×28×7=20+656=2656\frac{5 \times 4 + 3 \times 2}{8 \times 7} = \frac{20 + 6}{56} = \frac{26}{56}.

Establish the structural invariant and eliminate impossible options immediately.

STEP 2 • FAST SOLVE (LINE 2)15–25s
Line 2 (Halve): 1328\frac{13}{28}.

Evaluate the simplified expression using integer factoring or unit elimination.

Distractor Trap Forensics

The predictable cognitive traps engineered by examiners into the wrong options.

TRAP #1fence_post_or_domain_trap

Using denominator N instead of N-1 on the second draw in sampling without replacement

Treating dependent draws without replacement as independent draws with replacement.

Sub-90 Fix: Test invariants before expanding equations.
TRAP #2factor_of_two_or_stoichiometry_trap

Calculating P(A then B) = p(A) * p(B) without multiplying by 2 for ordered pairs (AB or BA)

Counting only one branch order when two distinct orderings satisfy the event.

Sub-90 Fix: Test invariants before expanding equations.
TRAP #3intermediate_stop_trap

Outputting P(fail all) = qⁿ instead of P(at least one success) = 1 - qⁿ

Forgetting to subtract the complementary probability from 1.

Sub-90 Fix: Test invariants before expanding equations.

Try 1 Timed Question: Multi-Stage Conditional Probability Trees (Without Replacement) & Algebraic Venn Overlaps

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Full Lesson All 15 Questions (ARCH-M1-12)
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ARCH-M1-12: Multi-Stage Conditional Probability Trees (Without Replacement) & Algebraic Venn Overlaps (Sample)Mathematics 1
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SUB-90 ZONE (<35S)
0.0s35s Sub-9090s ESAT Budget150s Classroom
Question 1 of 1ARCH-M1-12 (6-Card Lesson)Format: Standard 5-Option
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Question 1 of 1 • 90s Target

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