AMC 10 · 2008 · #17

Grade 7 probability

Archetype Overlap / Complement Counting └ Sub-archetype At-Least-One Complement: 1 − P(None)

probability-basiccombinations-basic systematic-enumeration ↑ Prerequisites: probability-basic

📏 Medium solution 💡 2 insights

Problem

70% of voters approve of the mayor. A pollster picks a voter at random three separate times, each pick independent of the others. Find the probability that exactly one of the three picked voters approves (so the other two do not).

Pick an answer.

(A)

0.063

(B)

0.189

(C)

0.233

(D)

0.333

(E)

0.441

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

'Exactly one approves' can happen in more than one way — the approver could be the first, second, or third voter. So the reliable move is to list every arrangement of one approve (Y) and two disapprove (N) among the three occasions (Tool #2): YNN, NYN, NNY. Each arrangement is a separate subproblem (Tool #7): find the probability of that one ordering, then add the arrangements together. Because the three picks are independent, each ordering has the same probability, so we compute it once and multiply by the number of arrangements.

1STEP 1

Turn the percent into probabilities

A random voter approves with probability 0.7, so the chance of not approving is 1 - 0.7 = 0.3, the same on every pick.

P(approve) = 0.7, P(not approve) = 1 - 0.7 = 0.3

2STEP 2

List the ways exactly one approves

Write Y for approve, N for not. The single Y sits in position 1, 2, or 3: YNN, NYN, NNY — 3 arrangements.

YNN, NYN, NNY → 3 arrangements

3STEP 3

Probability of one arrangement

Independent picks multiply: P(YNN) = 0.7 × 0.3 × 0.3 = 0.063, and every arrangement has one 0.7 and two 0.3's, so all match.

P(YNN) = 0.7 × 0.3 × 0.3 = 0.063

4STEP 4

Add the arrangements

The three arrangements cannot overlap, so add them: 3 × 0.063 = 0.189, which is choice (B).

3 × 0.063 = 0.189 → (B)

Answer

0.189

The number should be between 0 and 1, and 0.189 is. It also feels right that it is on the small side: approval is common (0.7), so getting only one approver out of three picks is a bit unlucky, which fits a modest probability. A quick trap check: (A) 0.063 is the probability of just one ordering (like YNN), forgetting there are three orderings; multiplying by 3 lifts it to 0.189, so (B) — not (A) — is the full answer. The four full cases also add to 1: no approvers 0.3³ = 0.027, exactly one 0.189, exactly two 3(0.7²)(0.3) = 0.441, all three 0.7³ = 0.343, and 0.027 + 0.189 + 0.441 + 0.343 = 1.000.

💡Key takeaway

For 'exactly one', first count the ways it can happen (here 3), find the chance of one way (0.7×0.3×0.3), then multiply.

Turn the percent into probabilities
List the ways exactly one approves
Probability of one arrangement
Add the arrangements

Turn the percent into probabilities

List the ways exactly one approves

Probability of one arrangement

Add the arrangements

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