Chance of the same worker getting X tasks out of Y total (in a pool of Z workers)

Question

This is basically a probability theory question, but I am so rusty that I can't seem to wrap my head around where to start.

I have a pool of Z workers. Each worker has a limit, X, for how many tasks it must perform before it is recycled. Each new task is grabbed by a worker at random (1/Z chance of a worker grabbing it).

After Y tasks has been given, what is the odds that one worker has reached the X threshold?

I wish to calculate it because I need to perform a "cleanup" periodically, and instead of picking some number at random, I would like it to be related to the number of workers and their task limit (which I may tweak at a later point), so I'll do the clean-up once there's a 25-30-35% chance that one worker is done for.

Answer 1

Consider worker W. Each time a task enters the system, there is a 1/Z chance that W grabs the task. A total of Y tasks enter the system.

Y tasks, each grabbed with probability 1/Z: the number of grabbed tasks, G_W, is a random variable with a binomial distribution.

The probability worker W has reached the X threshold after Y tasks is P(G_W >= X).

You can easily approximate the distribution and this probability by using a normal distribution with mean Y/Z and variance Y(Z-1)/Z^2. Then you can use a function that evaluates a normal distribution's cumulative distribution, which is available in mostly every programming language.