AMC 10 · 2016 · #19

Grade 8 probability

Archetype Branching State Enumeration └ Sub-archetype Markov State Probability

probability-basicindependent-eventsrecursive-sequencegeometric-series-infinite caseworkconvert-to-algebra ↑ Prerequisites: probability-basicindependent-events

📏 Medium solution 💡 3 insights

Problem

Each player flips until a first success and all must take the same number of tries. Find the probability.

Pick an answer.

(A)

$\frac{1}{8}$

(B)

$\frac{1}{7}$

(C)

$\frac{1}{6}$

(D)

$\frac{1}{4}$

(E)

$\frac{1}{3}$

How to solve

Strategy Introduce a Variable

There are infinitely many ways to win — all three stop on flip 1, or all on flip 2, or all on flip 3, and so on — so there is nothing finite to count. The way out is to stop tracking players one at a time and instead line the flips up into rounds that all three take together. Then one round has only eight possible results, and sorting those eight reveals that the whole game either finishes in round one or restarts from scratch. That repeat is the whole problem: name the unknown probability, let the restart bring the same unknown back on the right-hand side, and one linear equation replaces an infinite computation.

1STEP 1

Line the flips up in rounds

Rounds line the three players up.

all three flip n times ⇔ rounds 1,…,n-1 are all tails and round n is all heads

2STEP 2

List what one round can do

A round has eight equally likely outcomes.

P(one specific triple) = (1/2)³ = 1/8

3STEP 3

Sort the eight into three fates

They sort into three fates.

HHH: win 1/8, TTT: no change 1/8, 6 mixed: loss 6/8

4STEP 4

Spot the exact repeat

One fate repeats the situation exactly.

P(still undecided after n rounds) = (1/8)ⁿ ⟶ 0

5STEP 5

Name it and write one equation

That gives one self-referential equation.

P = 1/8 · 1 + 1/8 · P + 6/8 · 0 = 1/8 + 1/8P

6STEP 6

Solve for P

Solving gives 1/7, choice (C).

8P = 1 + P → 7P = 1 → P = 1/7

Answer

1/7

The size is right. Winning in round one alone already contributes 1/8 = 0.125, so the answer must be strictly larger than 1/8, which rules out (A). Everything beyond round one is reachable only through the single all-tails door of probability 1/8, so P = 1/8 + 1/8P with P ≤ 1 forces P ≤ 1/4, and feeding that bound back gives P ≤ 1/8 + 1/8 · 1/4 = 0.15625 — too small for (C) ≈ 0.167, (D) 0.25, and (E) ≈ 0.333. Only 1/7 ≈ 0.1429 survives. A second, sharper check comes from symmetry: ignore the all-tails rounds, which decide nothing, and look at the first round that does decide something. That round is equally likely to be any one of the 7 patterns other than TTT, and exactly one of those 7 is HHH, giving 1/7 straight away.

💡Key takeaway

When a situation can throw you right back to where you started, give the answer a name and let it show up on both sides of one equation.

Line the flips up in rounds
List what one round can do
Sort the eight into three fates
Spot the exact repeat
Name it and write one equation
Solve for P