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

A poll shows that $70\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work?

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: $70\%$ 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).

Givens: The probability a randomly picked voter approves is $70\% = 0.7$.; So the probability a randomly picked voter does **not** approve is $0.3$.; Three voters are picked independently, one on each occasion.; Answer choices: (A) 0.063, (B) 0.189, (C) 0.233, (D) 0.333, (E) 0.441.

Unknowns: The probability that exactly one of the three picked voters approves.

Understand

Plan

Primary tool: #2 Make a Systematic List

Secondary: #7 Identify Subproblems

'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.

Execute — Answer: B

#7 Identify Subproblems 6.RP.A.3 Step 1

Turn the percent into probabilities

Rewrite the poll figure as decimals.
A random voter approves with probability $0.7$, so the voter does not approve with probability $1 - 0.7 = 0.3$.
These two numbers are all we need, and they stay the same on every occasion because each pick is made at random from the same population.

$$P(\text{approve}) = 0.7,\qquad P(\text{not approve}) = 1 - 0.7 = 0.3$$

💡 A percent is just a fraction of the whole, so $70\%$ is the decimal $0.7$ and the rest, $0.3$, is everyone else.

#2 Make a Systematic List 7.SP.C.8 Step 2

List the ways exactly one approves

Let Y mean the picked voter approves and N mean they do not.
'Exactly one approves' means one Y and two N's spread across the three occasions.
List every placement of the single Y: it is either the first, second, or third pick.
That gives three arrangements — YNN, NYN, NNY — and no others.

$$\text{YNN},\ \text{NYN},\ \text{NNY} \;\Rightarrow\; 3\text{ arrangements}$$

💡 Choosing which one of the three occasions is the 'approve' occasion is the only choice to make, and there are three occasions.

#7 Identify Subproblems 5.NBT.B.7 Step 3

Probability of one arrangement

Take a single arrangement, say YNN: the first voter approves ($0.7$) and the next two do not ($0.3$ each).
Because the picks are independent, multiply the three probabilities.
Every arrangement has one $0.7$ and two $0.3$'s, so they all give the same product $0.063$.

$$P(\text{YNN}) = 0.7 \times 0.3 \times 0.3 = 0.063$$

💡 For independent events, the chance they all happen is the chances multiplied together.

#2 Make a Systematic List 7.SP.C.8 Step 4

Add the arrangements

The three arrangements are separate, non-overlapping ways to get exactly one approver, so add their probabilities.
Since each equals $0.063$, the total is $3 \times 0.063 = 0.189$.
That matches choice (B).

$$3 \times 0.063 = 0.189 \;\Rightarrow\; \textbf{(B)}$$

💡 Different ways of getting the same outcome can't happen at once, so their chances just add up.

[1] #7 6.RP.A.3 Rewrite the poll figure as decimals. A random voter approves with probability $0

[2] #2 7.SP.C.8 Let Y mean the picked voter approves and N mean they do not. 'Exactly one approv

[3] #7 5.NBT.B.7 Take a single arrangement, say YNN: the first voter approves ($0.7$) and the nex

[4] #2 7.SP.C.8 The three arrangements are separate, non-overlapping ways to get exactly one app

Review

Reasonableness: 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 $0.3^3 = 0.027$, exactly one $0.189$, exactly two $3(0.7^2)(0.3) = 0.441$, all three $0.7^3 = 0.343$, and $0.027 + 0.189 + 0.441 + 0.343 = 1.000$.

Alternative: Use the binomial formula directly. With $n = 3$ picks and success probability $p = 0.7$, the probability of exactly $k = 1$ success is $\binom{3}{1} p^1 (1-p)^2 = 3 \times 0.7 \times 0.3^2 = 3 \times 0.063 = 0.189$. The $\binom{3}{1} = 3$ counts the same three arrangements the list found, confirming (B).

CCSS standards used (min grade 7)

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Converting $70\%$ to the probability $0.7$ and its complement $0.3$.)
7.SP.C.8 Find probabilities of compound events using organized lists, tables, and simulation (Listing the three arrangements YNN, NYN, NNY of exactly one approver and adding their probabilities.)
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Multiplying $0.7 \times 0.3 \times 0.3 = 0.063$ and computing $3 \times 0.063 = 0.189$.)

⭐ For 'exactly one', first count the ways it can happen (here $3$), find the chance of one way ($0.7\times0.3\times0.3$), then multiply.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Turn the percent into probabilities

List the ways exactly one approves

Probability of one arrangement

Add the arrangements

Review

CCSS standards used (min grade 7)

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