The relationship of the genes and the chances for survival for the ptarmigan bird n is that the genes allow it to have two colours of feathers: white in the winter and brown in the summer.
This means that the bird will be less visible to predators in the winter, as it can hide in snow but also less visible to predators in the summer, as it can hide in grass!
As a result, more individuals will escape the predators and the species is more likely to survive.