AMC 8 · 2011 · #12

Grade 7 probability

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

probability-basicsystematic-enumerationpermutations-basic systematic-enumerationcasework ↑ Prerequisites: probability-basic

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Problem

Four people — Angie, Bridget, Carlos, and Diego — sit at random around a square table, one person per side. What is the probability that Angie and Carlos end up on opposite sides?

Pick an answer.

(A)

frac14

(B)

frac13

(C)

frac12

(D)

frac23

(E)

frac34

AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

Probability with equally likely outcomes is favorable/total, so the job is just careful counting. Tool #1 (Draw a Diagram) pins down what "opposite" means on a square — a seat has one opposite seat (across) and two adjacent seats. Tool #9 (Easier Related Problem) shrinks the sample space: by the rotational symmetry of the square we can fix Angie on one side, turning 4! = 24 arrangements into just 3! = 6 arrangements of the other three people. Tool #2 (Systematic List) then enumerates those 6 cases and counts how many put Carlos directly across from Angie.

1STEP 1

Label the square's sides Top, Right, Bottom, Left: Top faces Bottom, Left faces Right, so every side has exactly one opposite side.

opposite pairs = {Top, Bottom}, {Left, Right}

2STEP 2

Rotating the table changes nothing, so fix Angie at the Top; only the other three shuffle, leaving 3! = 6 arrangements.

4! = 24 → 3! = 6 arrangements

3STEP 3

List all 6 orderings of (Bridget, Carlos, Diego) into (Right, Bottom, Left), alphabetically by who takes the first seat.

(B,C,D), (B,D,C), (C,B,D), (C,D,B), (D,B,C), (D,C,B)

4STEP 4

Carlos faces Angie only in the Bottom (middle) slot: (B,C,D) and (D,C,B) — that's 2 of 6 cases.

P(Carlos opposite Angie) =

\frac{2}{6}

\frac{1}{3}

→ (B)

Answer

frac13

Quick sanity check by direct reasoning: after Angie picks any side, Carlos is equally likely to land on any of the 3 remaining sides. Exactly 1 of those 3 sides is opposite Angie, so P =

\frac{1}{3}

. This matches the listed count

\frac{2}{6}

\frac{1}{3}

and lands on choice (B). The answer also passes the gut-check: "opposite" is the strictest of the three positions Carlos can take (opposite, left of Angie, right of Angie), so a probability near

\frac{1}{3}

— not

\frac{1}{2}

or higher — is what we expect.

💡Key takeaway

Fix Angie at one side, list where the other three can sit, and count: this AMC 8 probability question is a clean Grade 7 "favorable over total" calculation.

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