Competition · AMC preparation · step 4 of 4
AMC 8 · 2004 · #21
Grade 7 probability
Pick an answer.
AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The event "product is even" breaks into three favorable cases (even×even, even×odd, odd×even), but its complement "product is odd" is a single case: both spinners land on an odd number. Tool #16 (Change Focus / Count the Complement) swaps the messy event for the clean one, computes the probability of the complement, and subtracts from 1. Tool #7 (Identify Subproblems) splits that complement into two independent subproblems — P(A odd) and P(B odd) — which multiply because the spinners are independent.
Flip to the complement
The product is odd only when both spins are odd, so P(even) = 1 - P(both odd) — one case, not three.
Even × anything is even, so the only way to get an odd product is two odd factors. The complement is much smaller, so count it instead.
The chance that the product is even equals one minus the chance that both spinners land on odd numbers.
▸ Why?
The product is odd exactly when both factors are odd, since odd times odd stays odd while an even factor makes the whole product even; so 'product even' is precisely the opposite case to both spinners landing on odd numbers.
▸ Why?
Every spin outcome makes the product either even or odd and never both, so the whole set of equally likely outcomes divides into just those two groups with no gaps or overlaps, and their chances add back to one.
Find the odd chance on spinner A
Spinner A: two of its four labels (1, 3) are odd, so P(A odd) = = .
Equally likely regions means favorable ÷ total. Two odd labels out of four gives one-half.
7.SP.C.7Identify SubproblemsFind the odd chance on spinner B
Spinner B: two of its three labels (1, 3) are odd, so P(B odd) = .
Same rule: two of the three labels are odd, so P = 2/3.
7.SP.C.7Identify SubproblemsMultiply and subtract from 1
Independent, so P(both odd) = × = , and the even product is the rest: 1 - = , choice (D).
Independent events multiply, so 1/2 × 2/3 = 1/3 for the odd-product side. The even-product probability is everything else, which is 2/3.
7.SP.C.8Change Focus Count The ComplementEven products have many sub-cases, but odd products only happen one way — both spinners must land on odd. Count that single easy case (), then take the complement: 1 - = .
- Flip to the complement
- Find the odd chance on spinner A
- Find the odd chance on spinner B
- Multiply and subtract from 1
A parent dashboard for the family lives at sensimlab.com.