AMC 10 · 2005 · #9

Grade 7 probability

Archetype Overlap / Complement Counting └ Sub-archetype Parity Partition: Complement via Odd-Even Split

probability-basicparity caseworkidentify-subproblems ↑ Prerequisites: probability-basicparity

📏 Medium solution 💡 2 insights

Problem

Fair die A has faces 1,1,2,2,3,3 and fair die B has faces 4,4,5,5,6,6. Both dice are rolled and the two top numbers are added. Find the probability that this sum is odd.

Pick an answer.

(A)

$\frac{1}{3}$

(B)

$\frac{4}{9}$

(C)

$\frac{1}{2}$

(D)

$\frac{5}{9}$

(E)

$\frac{2}{3}$

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

A sum is odd exactly when one addend is odd and the other is even, so Tool #7 (Identify Subproblems) splits the event into two clean, non-overlapping cases: (A odd, B even) and (A even, B odd). Tool #2 (Make a Systematic List) does the counting each case needs — tally how many of each die's six faces are odd versus even to get the single-die probabilities. Tool #3 (Eliminate Possibilities) is a quick backstop: since die A is odd more often than not while die B is even more often than not, an odd sum is a bit more likely than 1/2, which already points past choices (A), (B), and (C).

1STEP 1

Count odd and even faces

Sort the faces by parity: die A is odd on 4 of 6 faces (1,1,3,3), die B on only 2 of 6 (5,5).

P(A odd)=4/6=2/3, P(A even)=1/3; P(B odd)=2/6=1/3, P(B even)=4/6=2/3

2STEP 2

Split the odd sum into cases

A sum is odd only when the parities differ, so exactly two cases work: A odd with B even, or A even with B odd.

sum odd ⇔ (A odd, B even) or (A even, B odd)

3STEP 3

Find each case's probability

Independent rolls multiply: (A odd, B even) gives 2/3 × 2/3 = 4/9, and (A even, B odd) gives 1/3 × 1/3 = 1/9.

P(Case 1)=2/3·2/3=4/9, P(Case 2)=1/3·1/3=1/9

4STEP 4

Add the two cases

The cases cannot overlap, and the denominators already match, so just add: 4/9 + 1/9 = 5/9 — choice (D).

P(sum odd)=4/9+1/9=5/9 → (D)

Answer

5/9

The result 5/9 is a valid probability between 0 and 1, and it is just over 1/2 — sensible, since die A leans odd while die B leans even, making the mismatched (odd sum) pairing slightly favored. A full check confirms it: an even sum needs both odd (2/3·1/3=2/9) or both even (1/3·2/3=2/9), totaling 4/9, and 5/9+4/9=1 as it must. Choice (C) 1/2 is the trap for anyone who assumes odd and even sums are equally likely; here the loaded parities break that symmetry.

💡Key takeaway

A sum is odd only when exactly one die is odd, so multiply the matching odd/even chances for each case and add: 4/9+1/9=5/9.

Count odd and even faces
Split the odd sum into cases
Find each case's probability
Add the two cases

Count odd and even faces

Split the odd sum into cases

Find each case's probability

Add the two cases

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