Why does it matter if we're biased. I understand it would be not idea, but if we just had a large enough number, isn't it enough? lot of the time it seems like we've converged a lot before like 100?
I think our goal is always to make the algorithm as efficient as possible, so having a fixed large number such as 100 wouldn't be ideal, especially if the scene is pretty accurate to the final color we want after 10-15 bounces (which seems to be the case when I did some quick search but you can look into this). This will be unnecessarily expensive. And again, fixing the scene to a certain low number wouldn't work either (biased), which is why we use Russian Roulette.
In general, "bias" isn't necessarily a bad feature of statistical estimators, but subjectively, being unbiased simply feels better. Usually unbiased estimators are accepted if the bias is low or if the increased bias decreases overall MSE or some other loss; in this case, that's pretty hard to calculate so being unbiased is better.
So the goal is just to be unbiased. Even if we take a set number of trials, it's still biased (even if just slightly), so the entire image will be a little "off." The Russian Roulette idea will spread this bias around (some places will look more "off", others will look close to the original), making the whole image as whole hopefully look better.
I don't know if this is pedantic, but the professor says that Russian Roulette will not infinitely recurse. But isn't there a non zero probability that it could? Or more practically given enough rays there could be outliers of very large N bounces that act as bottlenecks?