Gene pool simulation
I am not sure I'm answering your question but I hope this will help.

There are two main models of genetic drift in biology: Moran model and Wright-Fisher model.

The Wright-Fisher model implies picking $N$ beads (where $N$ is the population size) with replacement in order to form the new population. Therefore, the change in allele frequency (due to genetic drift) follows a Poisson distribution.

$$\frac{2N!\cdot p^kq^{2N-k}}{k!(2N-k)!}$$

The Moran model consists of the following: At each time step you draw two beads. One disappear from the population and the other replicate in order to keep a stable population size.

The two models yield to similar results. Moran model runs twice as fast as Wright-Fisher model.

I don't fully understand your model. You say that in your model you pick one bead at a time. What do you do with this bead? How do you construct your population at the next time step? Do you allow changes in population size? Can you please try to develop how you defined your model. If it is a computer simulation you might paste the important part of your code, that might help.

Haldane and Kimura both worked a lot on these models of genetic drift. For example, they provide different ways of calculating the mean time to fixation of a given allele given their selection coefficient $s$ starting from a given initial frequency $p_0$. Also some work has been done on genetic drift in population where the population size changes over time. Some work have been performed considering more or less complex probability density function for the number of offspring per individual. This kind of literature is very interesting but sometimes ask for quite good knowledge in mathematics (especially for Kimura's articles).

I once asked this question concerning the different models of genetic drift and their assumptions but I didn't get an answer yet.