About a criteria for random integers number generation (C)

I am running a bunch of physical simulations in which I need random numbers. I'm using the standard rand() function in C++.

So it works like this: first I precalculate a bunch of probabilities that are of the form 1/(1+exp(a)), for a set of different a. They're of type double as returned by the exp function in the math library, and then things must happen with those probabilities, there are only two of them, so I generate a random number uniformly distributed between 0 and 1 and compared with those precalculated probabilities. To do that, I used:

double p = double(rand()%101)/100.0;

so I'm given random values between 0 and 1 both included. This didn't yield to correct physical results. I tried this:

double p = double(rand()%1000001)/1000000.0;

And this worked. I don't really understand why so I would like some criteria about how to do it. My intuition tells that if I do

double p = double(rand()%(N+1))/double(N);

with N big enough such that the smallest division (1/N) is much smaller than the smallest probability 1/1+exp(a) then I will be getting realistic random numbers.

I would like to understand why, though.