A walk through a challenging probability problem

Saw a great problem from the 2009 AMC 10 last night:

Here’s the problem:

Problem 24 from the 2009 AMC 10a

3 verticies of a cube are chosen at random.  What is the probability that the plane formed by those three verticies contains points inside of the cube?

This is a great problem (and a challenging one, too).   My older son took a few missteps walking through it, but our whole conversation about the problem this morning was great.

here’s the first part:

 

In the second part he takes a little misstep, but hopefully the mistake turned into a good learning opportunity:

Now we back up to where the misstep happened and try to get to the end of the problem:

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