There are two parts to this question.
The first is to recognize that in those four rolls, if any single number turned out to be even, then the product would be even. So instead of asking, "What's the probability that this turns out to be even?" let's turn it around into "What's the probability that this turns out to be odd?" The reason for this change is because there's only one way the product can be odd: if every number rolled is odd.
There are four possible rolls. On each roll, you have 3 possible outcomes that would give you an odd number (1, 3, 5). So, there are 3x3x3x3 possible combinations of number from the four rolls that could give you all odd numbers. 3^4 = 81.
In general there are 6x6x6x6 possible outcomes in total. 6^4 = 1296.
So the possibility of an odd product is 81/1296 or 1/15.
Since the only other outcome is an even product, that means there's a 15/16 chance of the product being even.
