Step-by-Step Lesson • Step 1 of 6

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

Step 1 of 6 • The Core Idea

ARCH-M1-12: Why the 140s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~140s (Blows 90s Clock)

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}

The Fast ShortcutTarget: 40s (Saves 100s)

Line 1 (Unordered pairs shortcut):

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

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Line 2 (Halve):

\frac{13}{28}

Where Students Lose Time: 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.

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