Question: I'm currently in the middle of a question where I'm given a 3x3 matrix:

$$\left(\begin{array}{rrr}
3 & 0 & 0\\
-2 & 7 & 0\\
4 & 8 & 1
\end{array}\right).$$

and have been asked to find the characteristic polynomial, eigenvalue and eigenvector/eigenspace when $\lambda = 3$

the aswer I got is $$ \left(\begin{array}{r}
\frac14\\
\frac18\\
1 
\end{array}\right) $$

The question then asks to find the basis of this eigenspace, how do I go about that?

Answer: 
If the probability is more than 0.5, expect the guest to be present in the most likely set. Otherwise, expect the guest to be absent.

This will maximize the likelihood.
For your edited question, the chance of 0 is .6^3 or .216. The flip side is a 79.4 chance of 1 or more guests. So the 1.2 projection is more accurate than defaulting to zero.