A mix between two pure bred dogs is called a designer dog. In the case of a Great Pyrenees Bernese Mountain dog mix, it is never 100 percent certain what a dog's temperament will be in any given circumstance.
Not all designer dogs are bred together at a 50-50 ratio. A lot of them have varying amounts of crossing, or even different breeds in their pedigree, that may not be obvious at first glance, especially if a certain trait was desired. The best thing to do with designer dogs is to research all the breeds that went into the cross and know that any one of them could result, and then prepare for each.