AMC 10 · 2006 · #17

Grade 7 probability

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

probability-basicratio-proportionsystematic-enumeration identify-subproblems ↑ Prerequisites: probability-basic

📏 Long solution 💡 3 insights

Problem

A die is weighted so each face's chance is proportional to its number. Two such dice are rolled. Find the probability the two faces add to seven.

Pick an answer.

(A)

$\frac 4{63}$

(B)

$\frac 18$

(C)

$\frac 8{63}$

(D)

$\frac 16$

(E)

$\frac 27$

How to solve

Strategy Introduce a Variable

The ratio 1:2:3:4:5:6 does not name the probabilities directly, so Tool #4 (Introduce a Variable) supplies the missing scale: call the smallest probability p, write the six probabilities as p,2p,…,6p, and use the fact that they add to 1 to pin down p. Once each face has a known probability, Tool #2 (Make a Systematic List) sweeps out every face pair that totals 7 so none is missed and none double-counted. Tool #7 (Identify Subproblems) splits the job cleanly: first find the single-die probabilities, then find each pair's probability by multiplying, then add the pieces — a sum-of-products that collapses to one fraction.

1STEP 1

Name the scale with a variable

A scale factor writes every face's chance with one letter.

P(k)=k p for k=1,2,3,4,5,6

2STEP 2

Use total probability to solve for p

The chances must add to one, which pins that scale.

p+2p+3p+4p+5p+6p=21p=1 → p=1/21, P(k)=k/21

3STEP 3

List every face pair that totals 7

Independence makes each pair a product of its two chances.

P(a,b)=a/21·b/21=a b/441

4STEP 4

Add the pair probabilities and simplify

Adding the pairs and reducing gives 8/63, choice (C).

P(sum=7)=(6+10+12+12+10+6)/441=56/441=8/63 → (C)

Answer

8/63

A quick sanity check on size: with a fair die, a total of 7 has probability 6/36=1/6≈0.167. Here 8/63≈0.127 is a bit smaller, which fits — the weighting pushes rolls toward high faces like 6, and a high roll on one die needs a low roll (probability only 1/21 or 2/21) on the other to still total 7, so sevens become slightly harder. The value also lands between the two closely spaced trap answers 4/63 and 8/63, and 4/63 is exactly half of ours — the number you would get by counting only three pairs instead of all six ordered pairs, a reminder to count both orders.

💡Key takeaway

Turn a ratio into probabilities by giving it a variable and making them add to 1; then a compound total is the sum of every ordered way to make it, each way being the two independent chances multiplied.

Name the scale with a variable
Use total probability to solve for p
List every face pair that totals 7
Add the pair probabilities and simplify

Name the scale with a variable

Use total probability to solve for p

List every face pair that totals 7

Add the pair probabilities and simplify

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