Random factor for simulating a Poisson distribution with a tick-tick simulation

I've got a program that simulates events happening. I have the following constraints:

• The distribution should approximate a Poisson distribution, i.e., independent events occurring at a fixed rate.
• At time 1000 ticks, the event should have happened 100 times (on average). (1000 and 100 are placeholders; the real numbers are determined experimentally from the real-world system I'm trying to model.)

I'm currently using code of the following form:

def tick():
    doMaintenance()
    while random() < 0.1:
        eventHappens()

I use a while instead of an if to simulate the idea that these events are independent. Otherwise, there would be no chance of two events occurring in the same tick. However, I suspect that means that the random() < 0.1 (where random returns a number in the half-open range [0.0, 1.0)) is slightly inaccurate. (It's okay that I'm quantizing the event occurrence time.)

Can somebody suggest the correct random() < f constant to use if I want (in the general case) a Poisson distribution such that at time t there will be event count e? I believe that such a constant f exists, but its derivation is not obvious to me.

I'm putting this in stackoverflow.com so I can conveniently talk in coding terms and because I'm using a tick-tick simulation which is more familiar to numerical simulation programmers than mathematicians. If this is something more appropriate in math.stackexchange.com, though, let me know.