AMC 10 · 2023 · #19

Grade 8 probability

Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Figure-N Growth Sequence

probability-basicparitygenerating-functionscombinations-basicpattern-recognition easier-related-problempattern-recognitioncomplementary-counting ↑ Prerequisites: probability-basicparitycombinations-basic

📏 Long solution 💡 4 insights

Problem

Drop 2023 distinguishable balls into three bins, each ball uniformly at random and independently. Find the answer choice closest to the probability that all three bins end up with an odd number of balls.

Pick an answer.

(A)

$\frac{2}{3}$

(B)

$\frac{3}{10}$

(C)

$\frac{1}{2}$

(D)

$\frac{1}{3}$

(E)

$\frac{1}{4}$

How to solve

Strategy Solve an Easier Related Problem

We cannot count 3²⁰²³ outcomes directly. Tool #9 (Easier Problem) — replace 2023 with a small odd number like n = 1, 3, 5 — lets us compute the probability by hand. Tool #5 (Pattern) reveals the answer approaches 1/4 very quickly. Tool #16 (Complement) gives the cleanest algebraic confirmation. Finally tool #3 (Eliminate) matches our limit to the closest of the multiple-choice values.

1STEP 1

The smallest case

The smallest case is impossible.

n = 1: P₁ = 0

2STEP 2

With three balls

Count the three-ball case directly.

n=3: P₃ = 3!/3³ = 6/27 = 2/9

3STEP 3

With five balls

Count the five-ball case too.

n=5: P₅ = 60/243 = 20/81 ≈ 0.2469

4STEP 4

Watch the trend

The values converge on one number.

P₁ = 0, P₃ = 2/9 ≈ 0.222, P₅ = 20/81 ≈ 0.2469 ↗ 1/4

5STEP 5

Write the general formula

A general formula confirms the limit.

P_n = 1/4 (1 - 1/(3^ n-1)) → 1/4

6STEP 6

Match the choice

The nearest choice is one quarter.

P₂₀₂₃ ≈ 1/4 → (E)

Answer

1/4

Three quick checks pass. (1) The small-case probabilities 0, 2/9, 20/81 all satisfy the formula 1/4(1 - 1/3ⁿ⁻¹): for n=3, 1/4(1 - 1/9) = 1/4·8/9 = 2/9 — matches. For n=5, 1/4(1 - 1/81) = 20/81 — matches. (2) The limit 1/4 also has a heuristic: of the 2³ = 8 parity patterns of the three bin counts, only one is (O,O,O), giving a baseline 1/8... but conditioning on the sum being odd halves the space, doubling to 1/4. (3) 0.25 is closer to all the offered choices than the runners-up 3/10=0.3 (gap 0.05) and 1/3≈ 0.333 (gap 0.083).

💡Key takeaway

This AMC 12 problem only needs Grade 8 exponent reasoning you already know — try n = 1, 3, 5 balls instead of 2023 to see the probability climb 0, 2/9, 20/81, … toward 1/4. The general formula 1/4(1 - 1/3ⁿ⁻¹) has a correction so tiny for n=2023 that the closest choice is (E) 1/4.