Why do so many people apply to so few PhD programs? Most of the schools that I've looked at for PhD programs in Computer Science have between a 9-15% acceptance rate.  I've noticed similar acceptance rates for other programs outside of computer science; yet almost everyone that I've ever spoken to has only applied to a handful of schools.  This just doesn't make sense to me.  

Let's just say that each school that I apply to has ~15% acceptance rate, then if I only apply to only 6 schools, I have roughly a 63% chance of getting accepted to at least one (1-probability of getting declined everywhere).  That doesn't seem horrible, but if I know I want to do a PhD, I would like to have much higher odds, plus most of the programs are closer to a 10% admissions rate.  I understand that applications are costly both monetarily and temporally speaking, but having a 37% chance of not getting into any programs is pretty bad.  Whereas if I apply to 14 schools, I have about a 10% chance of not getting accepted anywhere.  Can someone please explain the thought process behind this?
Although, I like your way of analytical thinking, some ouf your assumptions do not hold:


Selection procedures of PhD programs are not random. 
The selection procedure of one PhD program is not independent from other programs, as it is likely based on similar characteristics and markers in your CVs. (Maybe, there is conditional independence given 3, but that is problematic because the value of 3 is relative to the local distribution of 4.) 
Your application profile does not change. 
You do not know who your competitors are which may also vary between programs (and time). 


Therefore, I would assume that people use their applications as measures to test the unknown distribution of the accessibility of PhD programs. Unfortunately, this accessibility changes every term/year because of different populations of applicants. Furthermore, information about the specific demands and requirements of PhD programs is limited because it is time-consuming to look them up and sometimes they are simply hidden. That makes it reasonable to concentrate only on few programs and draw convenient samples. If applicants fail, it is easier to adapt, as your application costs were moderate, and to finally accept that there were better competitors and go on with life.