AMC 10 · 2007 · #16

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 a, b, c, d are each picked at random and independently from the whole numbers 0 through 2007. Find the probability that ad - bc comes out even.

Pick an answer.

(A)
$\ \frac 38$
(B)
$\ \frac 7{16}$
(C)
$\ \frac 12$
(D)
$\ \frac 9{16}$
(E)
$\ \frac 58$

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

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: 0 to 2007 holds 1004 evens and 1004 odds, so each of a, b, c, d is even with probability 1/2, independently.

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

Parity of each product

A product is odd only if both factors are odd, so ad is odd with probability 1/4 and even with probability 3/4; bc behaves the same.

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

When is the difference even?

ad - bc is even exactly when ad and bc share the same parity, so split into two disjoint cases: both products even, or both odd.

ad - bc even ⇔ (ad, bc) both even, or both odd
4STEP 4

Multiply and add the cases

Both products even: 3/4 · 3/4 = 9/16. Both odd: 1/4 · 1/4 = 1/16. Disjoint, so add: 9/16 + 1/16 = 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

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