2009 AMC 10A Problems/Problem 24
Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?
First of all, number of planes determined by any three vertices of a cube is ( surface, opposing parallel edges, points cut by three remote vertices). Among these planes only surfaces will not cut into the cube. Secondly, to choose three vertices randomly, the four vertices planes each will be chosen times, while the three vertices planes each will be chosen once. To conclude, the probability of a cutting in plane is = .
Minor LaTeX edit by Arcticturn
We will try to use symmetry as much as possible.
Pick the first vertex , its choice clearly does not influence anything.
Pick the second vertex . With probability vertices and have a common edge, with probability they are in opposite corners of the same face, and with probability they are in opposite corners of the cube. We will handle each of the cases separately.
In the first case, there are faces that contain the edge . In each of these faces there are other vertices. If one of these vertices is the third vertex , the entire triangle will be on a face. On the other hand, if is one of the two remaining vertices, the triangle will contain points inside the cube. Hence in this case the probability of choosing a good is .
In the second case, the triangle will not intersect the cube if point is one of the two points on the side that contains . Hence the probability of intersecting the inside of the cube is .
In the third case, already the diagonal contains points inside the cube, hence this case will be good regardless of the choice of .
Summing up all cases, the resulting probability is:
Solution 3 (Cheap solution, same approach as Solution 2)
This problem can be approached the same way, by picking vertices, but with a much faster and kind of cheap solution: Pick any vertex A and a face it touches. For vertex B, out of the remaining vertices, of them aren't on the same face as the one chosen for vertex A, so vertex C can be placed anywhere and the plane will no matter what be in the cube. Therefore, the probability of choosing a valid vertex B is .
There are ways to pick three vertices from eight total vertices; this is our denominator. In order to have three points inside the cube, they cannot be on the surface. Thus, we can use complementary probability.
There are ways to choose three points from the vertices of a single face. Since there are six faces, .
Thus, the probability of what we don't want is . Using complementary probability,
Solution 5 (Casework)
This problem is fairly simple. Start with points WLOG.
Case : You pick 2 points that are diagonally across from each other but still on the same face. This happens with probability . We notice that out of the possible outcomes include a point inside the cube. Therefore, the probability of this happening is .
Case : You pick 2 points that a directly across from each other. This happens with probability . We notice that out of the possible outcomes include a point inside the cube. Therefore, the probability of this happening is .
Case : You pick 2 points that is a space diagonal of the cube. This happens with probability . Clearly, all of the points contain a point inside the cube, so our probability is .
Adding these probabilities gives us
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