In a podcast with Sean Carroll and Liv Boeree they discuss a result from game theory that the optimal strategy in the face of incomplete information can require random decision making. For example, without more information (e.g. the robot that uses high frame rate video and fast movement to see what the human is picking and quickly pick what beats it), the strategy in rock-paper-scissors that cannot be dominated is to choose which one your going to do randomly with equal probability.
Now, in most of the examples above the source of the randomness is entirely from sexual reproduction (meiosis). The only example I can think off the top of my head of where this isn't the case is for crocodiles (maybe only some) where the sex is fixed by the temperature the egg is at during a certain stage of its development. In a sense, for them, sex selection is completely non-genetic but is done as a response to a stimulus.
My question is this: are there examples of genes that control development in a non-deterministic way (i.e. are there genes that by their presence will produce a trait with some probability greater than zero and less than nearly 1)?
I think genes that can be switched on and off in response to relevant environmental stimuli shouldn't count (e.g. the development into a queen for a honey bee). I'm just interested in whether there are genes that function in a highly random fashion and if natural selection has lead to such genes being produced (something like the presence of gene X causes trait Y with 30% probability, all other genes being equal). I add the qualifier "highly" to the randomness because these processes are ultimately always going to have some randomness from thermodynamics, and our cells go to a lot of effort to mitigate that randomness in most situations.