Competition · AMC preparation · step 4 of 4

AMC 8 · 2002 · #12

Grade 7 probability

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

probability-basicfraction-arithmeticcomplementary-counting complementary-countingidentify-subproblems ↑ Prerequisites: fraction-arithmetic

📏 Short solution 💡 1 insight

Problem

A spinner is split into three regions A, B, C. The arrow lands on A with probability

\frac{1}{3}

and on B with probability

\frac{1}{2}

. Find the probability that it lands on C.

Pick an answer.

(A)

$\frac{1}{12}$

(B)

$\frac{1}{6}$

(C)

$\frac{1}{5}$

(D)

$\frac{1}{3}$

(E)

$\frac{2}{5}$

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Change Focus / Count the Complement

Region C is exactly the part of the spinner that is not A and not B. Tool #16 (Count the Complement) names this directly: instead of asking "what is P(C)?", ask "what is left over after A and B take their shares of the whole?" Since all probabilities must add to 1, the leftover is P(C) = 1 - [P(A) + P(B)]. Tool #7 (Identify Subproblems) splits the arithmetic into two clean pieces: first add P(A) + P(B) with a common denominator, then subtract that sum from 1.

1STEP 1

Add the two probabilities

Add the two known probabilities over the common denominator 6: A and B together fill $\frac{5}{6}$ of the spinner.

P(A) + P(B) = 1/3 + 1/2 = 2/6 + 3/6 = 5/6

2STEP 2

Take the complement

The three probabilities sum to 1, so C is the leftover: 1 - $\frac{5}{6}$ = $\frac{1}{6}$ , choice (B).

P(C) = 1 - 5/6 = 6/6 - 5/6 = 1/6 → (B)

Answer

1/6

Check the sum:

\frac{1}{3}

\frac{1}{2}

\frac{1}{6}

\frac{2}{6}

\frac{3}{6}

\frac{1}{6}

\frac{6}{6}

= 1. The three probabilities add to 1, as required for the only possible outcomes of the spin. Size check: region C should be the smallest region because A (

\frac{1}{3}

) and B (

\frac{1}{2}

) already take most of the spinner. Indeed

\frac{1}{6}

is smaller than both

\frac{1}{3}

and

\frac{1}{2}

, which matches.

💡Key takeaway

The whole spinner has probability 1, and regions A and B already take $\frac{1}{3}$ + $\frac{1}{2}$ = $\frac{5}{6}$ . Region C is just the leftover: 1 - $\frac{5}{6}$ = $\frac{1}{6}$ , answer (B).

Add the two probabilities
Take the complement

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Add the two probabilities

Take the complement

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