AMC 10 · 2007 · #12

Grade 7 probability

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

parityprobability-basiccomplementary-counting casework ↑ Prerequisites: probability-basic

📏 Medium solution 💡 3 insights

Problem

Four integers are picked at random and independently from a long even-length range. Find the probability that a certain difference of products comes out even.

Pick an answer.

(A)

$\frac 38$

(B)

$\frac 7{16}$

(C)

$\frac 12$

(D)

$\frac 9{16}$

(E)

$\frac 58$

How to solve

Strategy Solve an Easier Related Problem

The actual values from 0 to 2007 do not matter for whether ad-bc is even; only the parity (even or odd) of each number matters. So Tool #9 (Solve an Easier Related Problem) shrinks a question about 2008 possible values down to a coin-flip question: each variable is just 'even' or 'odd' with probability 1/2. Tool #16 (Change Focus) then aims at the parity of the two products ad and bc instead of the numbers themselves, since a product is odd only when both of its factors are odd. Tool #7 (Identify Subproblems) splits the goal into two clean pieces — the chance each product is even and the chance the two products match in parity — which multiply and add to the final probability.

1STEP 1

Shrink to a parity question

Only parity matters, so each pick is an even coin flip.

P(even) = 1004/2008 = 1/2, P(odd) = 1/2

2STEP 2

Parity of each product

A product is odd only when both factors are odd.

P(ad odd) = 1/2·1/2 = 1/4, P(ad even) = 3/4; same for bc

3STEP 3

When is the difference even?

The difference is even exactly when the two parities match.

ad - bc even ⇔ (ad, bc) both even, or both odd

4STEP 4

Multiply and add the cases

Adding the two matching cases gives 5/8, choice (E).

3/4·3/4 + 1/4·1/4 = 9/16 + 1/16 = 10/16 = 5/8 → (E)

Answer

5/8

The answer 5/8 = 0.625 is a valid probability (between 0 and 1) and sits just above 1/2, which fits: 'even' products are far more common than 'odd' ones (each product is even 3/4 of the time), so the two products agreeing in parity should be a bit more likely than a coin flip. A cross-check by complement confirms it: ad-bc is odd when the products differ in parity, with probability 3/4·1/4 + 1/4·3/4 = 3/16+3/16 = 6/16 = 3/8, and 5/8 + 3/8 = 1 as it must. Choice (A) 3/8 is exactly this complement (the 'odd' probability), a natural trap; (C) 1/2 would be right only if products were equally likely even or odd, which they are not.

💡Key takeaway

Forget the real numbers and track only even-or-odd: a difference is even when both products match in parity, and since a product is even 3/4 of the time, the chances stack to 5/8.

Shrink to a parity question
Parity of each product
When is the difference even?
Multiply and add the cases

Shrink to a parity question

Parity of each product

When is the difference even?

Multiply and add the cases

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