Defintely one to wonder about - not a single part of that statement is scientific, but it's entirely based on a particular slant on theology. In other words her definition of "random" has absolutely nothing to do with science, and ought to be clearly labelled "my theological opinion".
Less problems are caused by taking seriously this scientific definition of randomness:
[S]cientists use chance, or randomness, to mean that when physical causes can result in any of several outcomes, we cannot predict what the outcome will be in any particular case. (Futuyma 2005: 225)
In philosophical terms, Futuyma correctly treats randomess in science as exclusively epistemological (what we can know) whereas Applegate's definition is entirely ontological (what the true cause is).
Applegate's involves some contradictions - firstly according to Futuyma, we don't know the cause of random events through science; yet Applegate tells us the cause - a creation that can create itself, though unintelligently. Whence comes that knowledge?
Secondly, there are theological and philosophical problems with her statement: philsophically, for something to cause itself is absurd. Theologically, what does it mean for a "good creation" to be left to create, unintelligently, harmful things - and what does it actually mean (if anything) to invoke God's providence in that?
Thirdly, this "capability to create itself" appears to mean nothing more than "ontological randomness", that is, the stuff happens not just unintelligently, but for no actual reason at all. Now if one were to say that I have the "capability", every now and then, to do something entirely without cause - like punching staff in supermarkets, or jumping off buildings - you would say that's not a capability, but an affliction. The autonomy of madness is no freedom at all. And for the irrational creation to act mad is even less than that - it's God himself relinquishing control to insanity, if that were possible... it certainly doesn't sound "good".
Me, I prefer intelligently designed water - as you appear wisely to do!