To fully interpret a probability distribution you need to use integrals. For example, if I have a probability distribution over the number of heads in 50 coinflips and I want to know the probability that the observed value is going to fall within a certain interval, I have to take the integral of that part of the distribution. You can definitely understand what a probability distribution is without calculus, but you're going to have a hard time actually doing the math.

Edit: It occurs to me that statistical software could do most of the number-crunching for you, which would definitely make things easier.