Our planet is unusual in that the shape of the moon appears to fit
perfectly within the shape of the sun during a total eclipse, forming
the  beautiful phenomenon called the "string of pearls."  Can someone
break down statistically for me the likliehood of there being other
planets in other solar systems that have moons of the right size and
distance that bring about the same phenomenon when an eclipse occurs
for them?

This is a great question, but I don't think it can be answered with any precision.

"The problem is you need to know the joint probablity distribution of
satellite size and satellite-planet distance as a function of star
size and star-planet distance. Which we don't have."

http://www2b.abc.net.au/science/k2/stn/archives/archive34/newposts/256/topic256326.shtm

"One of the most startling facts about them is that they occur at all.
The moon, although 400 times smaller than the sun, is 400 times closer
and hence, from Earth, the lunar disc perfectly shrouds the star that
warms our solar system. The odds of such a coincidence occurring where
there is conscious life to observe it are longer than any number a
mathematician could conjure. So perfect is the match between the two
celestial bodies that in the last instant before the sun is obscured,
its rays shine through the gaps between the mountains on the moon's
edge, creating a breathtaking effect known as Baily's Beads."

http://www.smh.com.au/articles/2002/11/22/1037697867672.html

<<So perfect is the match between the two
celestial bodies that in the last instant before the sun is obscured,
its rays shine through the gaps between the mountains on the moon's
edge, creating a breathtaking effect known as Baily's Beads.>>

That's one of the quotes I was looking for.  This is a fact I find
absolutely amazing!  Thank you.