AMC 8 · 2007 · #21

Grade 7 probability

Archetype Overlap / Complement Counting └ Sub-archetype Two-Set Inclusion-Exclusion: Solve for Missing Region

probability-basiccombinations-basiccomplementary-counting caseworkcomplementary-counting ↑ Prerequisites: probability-basiccombinations-basic

📏 Short solution 💡 2 insights

Problem

A deck has 8 cards: red A, B, C, D and green A, B, C, D. Two cards are dealt at random. A "winning pair" means the two cards share a color or share a letter. Find the probability of a winning pair.

Pick an answer.

(A)

$\frac{2}{7}$

(B)

$\frac{3}{8}$

(C)

$\frac{1}{2}$

(D)

$\frac{4}{7}$

(E)

$\frac{5}{8}$

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Count Systematically

Probability with a small deck is a Grade 7 counting question: (favorable outcomes) ÷ (total outcomes). Tool #13 (Count Systematically) fits because we can anchor on the first card and count how many of the remaining 7 cards win with it. That "anchor and count" works for every starting card because of symmetry. As a check, Tool #7 (Split into Subproblems) breaks the winning event into two disjoint cases — "same letter" and "same color" — and adds their probabilities.

1STEP 1

Anchor on the first card: the 7 cards left are all equally likely, so the probability is (winning second cards) ÷ 7.

P(winning pair) = (winning second cards)/7

2STEP 2

If the first card is red, the other 3 red cards share its color — that's 3 same-color matches.

#{same color} = 3

3STEP 3

The only other card with the same letter is the green one — that's 1 same-letter match, and it's a different card.

#{same letter} = 1

4STEP 4

The two groups never overlap, so total winners = 3 + 1 = 4, and the probability is 4 out of 7.

P(winning pair) =

\frac{3 + 1}{7}

\frac{4}{7}

→ (D)

Answer

\frac{4}{7}

Cross-check by counting unordered pairs. Total pairs from 8 cards: C(8, 2) = 28. Same-color pairs: C(4, 2) + C(4, 2) = 6 + 6 = 12. Same-letter pairs: one pair per letter, 4 letters, so 4. These sets do not overlap (two distinct cards cannot share both color and letter), so winners = 12 + 4 = 16. Probability =

\frac{16}{28}

\frac{4}{7}

, matching (D). Also

\frac{4}{7}

≈ 0.57, a bit above one-half — reasonable since each card already has 4 partners out of 7.

💡Key takeaway

Anchor on the first card, then count the winners among the 7 left: 3 share the color, 1 shares the letter, total 4 out of 7.

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