75-Archetype SyllabusMathematics 1 • ARCH-M1-12 • Spec M7

Full 6-Card Lesson 15-Question Timed Drill

UAT-UK SPEC CODE: M7 (Probability)ASSUMED MATHS 1 PREREQUISITE: M7.7FREQUENCY: VERY HIGH

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

Deconstructed from 20 past UAT-UK questions (8.89% of all historical Mathematics 1 papers). Students using standard A-Level classroom methods spend an average of 140s on this archetype—blowing past the 90s ESAT budget. Using the 2-line Sub-90 heuristic below compresses solve time to 40s (saving 100s per question).

Historical Frequency

20 Past Qs

8.89% of Mathematics 1

Classroom Trap Time

140s

+50s over 90s budget

Sub-90 Target Time

40s

Banks +100s on the clock

Dominant UI Format

5-Option A–E

Maths 1 Skill: M7.7

Card 1 & Card 2 Preview: The Classroom Time Trap vs. Sub-90 2-Line Rule

Excerpted from our interactive 6-Step Topic Lesson for ARCH-M1-12.

Launch All 6 Active-Recall Cards

CARD 1 OF 6 • THE CLASSROOM TIME TRAP (140s)

Why Students Bleed Time:

Step 1: Identify mutually exclusive branches for 'same colour'.
The two counters are the same colour if they are both red (RR) or both blue (BB).

Step 2: Multiply probabilities along each branch without replacement.

P(\text{RR}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56}

P(\text{BB}) = \frac{3}{8} \times \frac{2}{7} = \frac{6}{56}

Step 3: Sum and simplify.

P(\text{same}) = \frac{20 + 6}{56} = \frac{26}{56} = \frac{13}{28}

In 20 past UAT-UK questions (8.89% of Mathematics 1), candidates lose 100s by treating dependent draws without replacement as independent draws with replacement. Instead, apply M7 with Maths 1 skill M7.7.

CARD 2 OF 6 • THE 2-LINE SUB-90 HEURISTIC (<40s)

Line 1 (Unordered pairs shortcut):

\frac{5 \times 4 + 3 \times 2}{8 \times 7} = \frac{20 + 6}{56} = \frac{26}{56}

Line 2 (Halve):

\frac{13}{28}

Sanity & Limit Check: Verify units/signs against fence_post_or_domain_trap (Using denominator N instead of N-1 on the second draw in sampling without replacement) and factor_of_two_or_stoichiometry_trap (Calculating P(A then B) = p_A * p_B without multiplying by 2 for ordered pairs (AB or BA)).

Examiner Distractor Blueprint for ARCH-M1-12

FOIL #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.

Open Fence-Post / Domain Trap Clinic

FOIL #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.

Open Factor-of-2 / Stoichiometry Clinic

FOIL #3intermediate_stop_trap

Outputting P(fail all) = q^n instead of P(at least one success) = 1 - q^n

Forgetting to subtract the complementary probability from 1.

Open Intermediate Stop Trap Clinic

FOIL #4subsystem_isolation_trap

Dividing by total population N instead of the conditioned subgroup size n(B) in P(A | B)

Using the unconditioned sample space as the denominator.

Open Subsystem Isolation Trap Clinic

Try 1 Live Sample Question Under the 90-Second Stopwatch

Press W to open the 2-line dry-wipe scratchpad, select A–E, and lock in to reveal the Dual-View solution.

Full 6-Card Lesson Launch All 15 Questions (ARCH-M1-12)

ARCH-M1-12: Multi-Stage Conditional Probability Trees (Without Replacement) & Algebraic Venn Overlaps (Sample)Mathematics 1

00:00.0

SUB-90 ZONE (<40S)

0.0s40s Sub-9090s ESAT Budget150s Classroom

Question 1 of 1ARCH-M1-12 (6-Card Lesson)Format: Standard 5-Option

Right-click any option to cross out • Keys A–E + Enter

Question 1 of 1 • 90s Target

Ready to Start Question 1?

The question stem, diagram, and answer options are hidden behind this pre-start gate. When you click below, the question will reveal and your 90-second split stopwatch will immediately begin ticking.

Mathematics 1•ARCH-M1-12•Standard 5-Option

or press Enter to start