6.13.  This is geometrical probability in 3-D.  The sample space is the unit cube, and we want $x+y>z$, $y+z>x$ and $z+x>y$.  You need to figure out the volume of the region of points satisfying all three inequalities.

6.17: This is similar to something we did before.  $x,y\in[0,1]$ and $|x-y|<1/3$.

6.19: To find the distribution, you need to compute the probability for each possible outcome.  The outcomes here are the number of times the bird needs to get out.  So you can find the probability that the bird gets out in one time, and the probability that the bird gets out in two tries, and so on, find the formula if $k$ tries are needed.  For part (b), you only need to calculate the probability for $1$, $2$, and $3$.

6.20: There is no shortcuts.  For part (a) you need to calculate the probability that they both hit 0 times, both 1 time, both 2 times, both 3 times.  Similar ideas for part (b).

The answer key is available on the Areteem Textbook Page.  You can find the link on the front page of this website.