clearly one step will take the ant to the equator of the octahedron.
From there, there are 4 ways to go, only one of which continues on to the bottom.
So, p = 1/4
An ant begins at the top of the pictured octahedron. If the ant takes two "steps", what is the probability it ends up at the bottom of the octahedron? Assume a "step" is a journey from one vertex to an adjacent vertex along an edge.
5 answers
2/3
An ant begins at the top of the pictured octahedron. If the ant takes two "steps", what is the probability it ends up at the bottom of the octahedron? Assume a "step" is a journey from one vertex to an adjacent vertex along an edge.
[asy]
import three;
triple A=(0,0,0),B=(2,0,0),C=(2,2,0),D=(0,2,0), EE = (1,1,2), F = (1,1,-2);
currentprojection=perspective(1/2,-1,1/2);
draw(A--B--C);
dot(EE);
draw(D--A,dotted);
draw(D--EE,dotted);
draw(C--D,dotted);
draw(A--EE--B);
draw(C--EE);
draw(A--F--B);
draw(C--F--D,dotted);
[/asy]
[asy]
import three;
triple A=(0,0,0),B=(2,0,0),C=(2,2,0),D=(0,2,0), EE = (1,1,2), F = (1,1,-2);
currentprojection=perspective(1/2,-1,1/2);
draw(A--B--C);
dot(EE);
draw(D--A,dotted);
draw(D--EE,dotted);
draw(C--D,dotted);
draw(A--EE--B);
draw(C--EE);
draw(A--F--B);
draw(C--F--D,dotted);
[/asy]
Verified Answer --- 1/4
Solution:
The ant can only move along adjacent edges, so he only has 4 ways to get to the bottom. He goes from the top to any vertex of the square, and then he moves to the very bottom vertex. Then, we have to find the total number of ways he can move in 2 steps. We already have the 4 that are correct, so don't forget them. The ant can move to any vertex on the square, and then move to the side both ways. He can do this on any edge, giving us 2*4 = 8 total ways to move on the edges of the square. Additionally, he can go down from the top, and then return to the top. this gives us 4 more ways. Adding these all together, we get
4 + 8 + 4 = 16. If we put our correct solutions over our total ways, we get 1/4.
Solution:
The ant can only move along adjacent edges, so he only has 4 ways to get to the bottom. He goes from the top to any vertex of the square, and then he moves to the very bottom vertex. Then, we have to find the total number of ways he can move in 2 steps. We already have the 4 that are correct, so don't forget them. The ant can move to any vertex on the square, and then move to the side both ways. He can do this on any edge, giving us 2*4 = 8 total ways to move on the edges of the square. Additionally, he can go down from the top, and then return to the top. this gives us 4 more ways. Adding these all together, we get
4 + 8 + 4 = 16. If we put our correct solutions over our total ways, we get 1/4.
The answer is 1/4 because the ant has 4 different ways to move the first step and 4 different ways to move the second step. This makes a grand total of 16 steps. Out of these only 4 will get the ant to its destination: the bottom of the octahedron. So, the probability is 4/16 = 1/4.
1 answer