AMC 10 · 2016 · #13

Grade 7 probability
Archetype Overlap / Complement Counting └ Sub-archetype Complement by Subtracting Forbidden Arrangements
probability-basicratio-proportionfloor-function bound-inequality-then-enumerateconvert-to-algebra ↑ Prerequisites: probability-basic
📏 Long solution 💡 4 insights
Problem
One marked item lands at random among many and a lopsidedness condition must fail. Add the digits of the smallest size.

Pick an answer.

(A)
12
(B)
14
(C)
16
(D)
18
(E)
20
How to solve
Strategy Change Focus / Count the Complement

The green balls are identical, so a random line-up is decided entirely by where the red ball slips in. Tool #16 (Change Focus) makes this the model: lay down the N green balls, and the red ball lands in one of the N+1 gaps, each gap equally likely. Now P(N) is just (favorable gaps)/(N+1). Tool #4 (Introduce a Variable) writes N=5k so the threshold 3/5N=3k is a clean whole number and the gaps are easy to count. Tool #13 (Convert to Algebra) turns P(N) ＜ 321/400 into an inequality in k, which solves to a smallest k; from there N=5k and the digit sum follow.

1STEP 1

Reframe as dropping the red ball in a gap

Only the gap matters, not the order.

outcomes=N+1, L∈{0,1,…,N}, right pile=N-L
2STEP 2

Count the gaps that work

Counting the good gaps is straightforward.

L ≥ 3k: 2k+1 gaps; L ≤ 2k: 2k+1 gaps; favorable=4k+2
3STEP 3

Write the probability and sanity-check it

The given endpoints check the formula.

P(N)=(4k+2)/(5k+1); k=1→6/6=1; k→∞→4/5
4STEP 4

Solve the inequality for k

The condition is one plain inequality.

400(4k+2) ＜ 321(5k+1)→ 479 ＜ 5k→ k ＞ 95.8→ k=96
5STEP 5

Find N and add its digits

Its digits add to 12, choice (A).

N=5 · 96=480, 4+8+0=12→(A)
Answer
12
The formula P(N)=(4k+2)/(5k+1) matches both anchors the problem hands you: P(5)=1 and the limit 4/5, so the count of favorable gaps is trustworthy. Since P(N) decreases toward 4/5=0.8 and 321/400=0.8025 sits just above 0.8, the crossing happens at a large N, which fits N=480. Checking the boundary: at k=95, 382/476=0.8025… which is not below 321/400, while at k=96, 386/481≈ 0.8025 ＜ 0.8025 holds, so N=480 is genuinely the first one to dip under.
💡Key takeaway

Since the green balls are identical, only the red ball's gap matters; count the gaps that leave a big enough pile, write the probability as a fraction in k, and solve the inequality to land on N=480 and digit sum 12.

  • Reframe as dropping the red ball in a gap
  • Count the gaps that work
  • Write the probability and sanity-check it
  • Solve the inequality for k
  • Find N and add its digits

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