Infinite Triangular Lattice

Probability Level 2

You are on an infinite triangular lattice with black circles repeated as shown below, and you start at the vertex circled in red:

Every move you randomly walk along a black line segment to a neighboring vertex.

What is the expected value for the number of moves before you hit one of the lattice points with a black circle on it?

1 2 3 4 5 6 7 Infinite

2 solutions

Geoff Pilling
Jun 22, 2016

Relevant wiki: Expected Value - Problem Solving

Every move you have $\frac{1}{3}$ chance of hitting a black circle, or $\frac{2}{3}$ chance of winding up effectively where you started (in between two on an infinite lattice).

So the expected number of moves is:

$E = 1 + 2/3(E)$

Or $E = \boxed{3}$

Miksu Rankaviita
Jun 23, 2016

Every vertex that doesn't have black circle on them, are surrounded by six vertecies. Two of them have black circle on them and the other four doesn't. This means that we have $\frac{2}{6}$ propability to hit vertex with black square so we will hit a vertex with a black cicle once in three steps so three is the right answer

Infinite Triangular Lattice

2 solutions

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