A one-liner.

Probably wouldn't win any prizes for efficiency, but scrunched up, it might come in handy in a golf-tournement someday.

my $choice = do {@_ = (); @_ = (@_, map{ ($_->[0]) x $_->[1] } [$a,$b]
+) while ($a,$b)= each %hash; $_[rand @_]};
[download]

And to prove it works, a one-line test suit.

perl -e"%h=(a=>200,b=>20,c=>2);$t{do{@_=();@_=(@_,map{($_->[0])x$_->[1
+]}[$a,$b]) while($a,$b)=each%h;$_[rand@_]}}++ for 1..1000;print%t;"

a901b86c13
[download]

Note: I added an unnecessary space to prevent PM from wrapping it at a less convenient place. It (just) fits on one line on my command line.