Randomness in living systems
In an evolution mutations are often random and lead to differences in phenotype that can be adaptive under certain pressures.  A lot of times mutation is a random process, but here are three cases I can think of off of the top of my head where I would say the organism is 'trying' to do it:

HIV is a retrovirus, which means in its viral form its genome is single stranded RNA, which is then converted into double stranded DNA within the host.  That conversion is carried out by a virally encoded reverse transcriptase.  This enzyme has a much higher error rate when making the RNA into DNA because it cannot proofread like our DNA polymerases.  This means that HIV mutates incredibly fast.  Many of these mutants are not very fit, but since there is so much selective pressure against staying the same due to attack from the immune system, some mutants will be much more fit if they can avoid that attack.  Its a numbers game, and by making lots of random mutants HIV is very good at it.

A related, but different case is found in the diversity generating retroelements of certain bacteriophages.  These are double stranded DNA viruses that infect bacteria.  The bacteria these viruses infect could mutate to escape the viruses by losing a certain receptor that the virus binds to.  It was observed however that the the virus would mutate incredibly quickly to bind to a different receptor, a lot faster than would be expected for a DNA virus.  Also, it was found that these viruses contained reverse transcriptases, which is bizarre, since there is no reverse transcription step in a dsDNA viral life-cycle (so we thought).  To make a long story short, this virus will transcribe the DNA that codes for its binding proteins into RNA, then use its reverse transcriptase to turn that RNA BACK into DNA, but uses a sophisticated targeting strategy so that it only mutates the region that is used for binding to the bacteria.  It will only mutate adenine residues, leaving the C's, T's and G's alone.  This is a much more clever system than HIV, because this virus uses site-specific mutations on its binding feet, and doesn't mutate its core proteins, the mutation of which would probably just lead to viruses that couldnt replicate.  Here a reference for the interested: http://www.nature.com/nature/journal/v431/n7007/abs/nature02833.html

Just so you dont think this is reserved to viruses, we do this as well with our immune system.  In order to recognize antigens in attacking pathogens, our immune cells have two methods for making diverse, random, receptors.  First is V(D)J recombination, where the multiple diverse copies of the V, D and J regions of antibodies and receptors are combined at random to make one set.  Wiki says theres 3x10^11 possible combinations here.  After that, theres also somatic hypermutation during the proliferation of B cells.  In this process, the random mutation rate is hugely increased in the region of the B cell receptor gene, specifically for making slightly different versions of the already 'OK' receptor to make it a great receptor.

General theme: raising your mutation rate, either across the board(HIV) or specifically (DGRE and SHM) is a good way to intentionally add randomness for a beneficial purpose (to the extent that you can attribute 'intent' to a viral particle).