AMC 10 · 2008 · #20
Grade 7 probabilityPick an answer.
AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The clean way through a two-dice problem is a full addition table (tool #2): rows are Die A's six faces, columns are Die B's six faces, and each cell holds the sum. That lays out all 36 equally likely outcomes at once so nothing is missed or double-counted. Tool #7 (Identify Subproblems) then splits the count into three easy tallies — how many cells equal 5, how many equal 7, how many equal 9 — and adds them. Tool #3 (Eliminate Possibilities) matches the reduced fraction to a choice and rules out the decoys. Tool #16 (Count the Complement) gives a fast independent sanity check via the complementary sums.
Set up the equally likely outcomes
Both dice are fair and independent, so all 6 × 6 = 36 face pairs are equally likely — keep the repeated faces as separate outcomes.
Fair and independent rolls make every face pair equally likely, so probability is just favorable pairs over 36.
Fair independent rolls make every face pair equally likely, so probability is favourable pairs over all pairs.
▸ Why?
No pair is favoured over another, so counting is enough to settle the chance.
▸ Why?
Each die is rolled without regard to the other, so the face counts multiply into the total.
Build the addition table
Put Die A's faces down the side and Die B's across the top, then fill each cell with its row-plus-column sum — all 36 sums at once.
A row-by-column table lists every compound outcome exactly once.
7.SP.C.8Make A Systematic ListCount the favorable cells
Sweep the table once per target: sum 5 fills 4 cells, sum 7 fills 6, sum 9 fills 4, so the favorable count is 14.
Breaking one big count into three target-sum counts keeps the tally error-proof.
7.SP.C.8Identify SubproblemsForm and reduce the probability
The probability is favorable over total, 14/36; dividing top and bottom by the common factor 2 gives 7/18.
Favorable over total gives the probability, and dividing by a common factor keeps its value unchanged.
4.NF.A.1Identify SubproblemsMatch the answer choice
7/18 is choice (B): 11/18 is the complement, 5/18 drops the sum-7 cells, and 3/4, 8/9 are far too large.
A valid probability sits between 0 and 1, and only 7/18 matches the honest count.
7.SP.C.5Eliminate PossibilitiesMake an addition table of all 36 face pairs, count the 14 cells that add to 5, 7, or 9, and the probability is 14/36 = 7/18.
- Set up the equally likely outcomes
- Build the addition table
- Count the favorable cells
- Form and reduce the probability
- Match the answer choice