AMC 10 · 2020 · #11

Grade 7 probability

Archetype Branching State Enumeration └ Sub-archetype Markov State Probability

probability-basicsymmetry-argumentcoordinate-geometryrecursive-sequencesystems-of-equations symmetry-argumentcaseworkidentify-subproblems ↑ Prerequisites: probability-basiccoordinate-geometry

📏 Long solution 💡 3 insights

Problem

A frog starts at the point (1, 2) inside the square with corners at the origin and (4,4). At each step it jumps one unit up, down, left, or right with equal probability, and it stops the first time it lands on a side of the square. Find the probability that the stopping side is a vertical one.

Pick an answer.

(A)

$\frac12$

(B)

$\frac 58$

(C)

$\frac 23$

(D)

$\frac34$

(E)

$\frac 78$

How to solve

Strategy Draw a Diagram

Tool #1 (Draw): sketch the 5 × 5 grid of lattice points and mark the start (1, 2) and the four boundary sides. Tool #15 (Reorganize): exploit the square's reflection symmetries — swapping x ⇔ y turns vertical sides into horizontal ones, so P_V(x,y) = 1 - P_V(y,x). Tool #7 (Subproblems): one jump from (1,2) goes to one of (0,2), (2,2), (1,1), (1,3); compute the four sub-probabilities, then combine with the law of total probability. Tool #3 (Eliminate): match the final fraction to the five choices.

1STEP 1

Draw the board and name the target

The target is stopping on a vertical side.

start = (1, 2), vertical sides: x = 0, x = 4

2STEP 2

Points on the diagonal

On the diagonal it is exactly one half.

P_V(x,x) = 1/2 → P_V(1,1) = P_V(2,2) = 1/2

3STEP 3

Carry values by symmetry

Reflecting gives the other point at once.

P_V(1, 3) = 1 - P_V(3, 1) = 1 - 1/2 = 1/2

4STEP 4

Write the start's equation

The start is the average of four neighbours.

P(1,2) = 1/4[P(0,2) + P(2,2) + P(1,3) + P(1,1)]

5STEP 5

Fill in the neighbours

One neighbour already sits on a side.

P(0,2) = 1, P(2,2) = P(1,3) = P(1,1) = 1/2

6STEP 6

Take the average

Average the four values.

P(1,2) = 1/4 · 5/2 = 5/8

7STEP 7

Match the choice

The probability is five eighths.

5/8 → (B)

Answer

5/8

The starting point (1, 2) is exactly 1 step from the nearest vertical side and 2 steps from the nearest horizontal side, so the answer should beat 1/2 (vertical is favored) but not by a huge amount. The exact value 5/8 = 0.625 is just above 1/2 — a moderate vertical advantage, exactly what closeness to one vertical side suggests.

💡Key takeaway

This AMC 12 problem only needs Grade 7 probability you already know! Reflecting the square across the line y = x swaps vertical and horizontal sides, which forces P = 1/2 at every diagonal point like (1, 1), (2, 2), (1, 3). One step from (1, 2) gives four equally likely cases: (0, 2) wins immediately (P = 1), and the other three land on diagonals (P = 1/2 each). Average them: 1/4(1 + 3/2) = 5/8, answer (B).