For the theoretical one, it's just a binomial distribution with 2000 trials, .04 success rate and you want the probability that less than 100 survive. There are a lot of calculators for this (http://stattrek.com/online-calculator/binomial.aspx or Excel), but I'm not sure of the level of detail you're expected to have.
Let X be the number of insects that survive (a random variable).
1. Straight up calculating P(X=0) + P(X=1) + ... + P(X=100) is taxing, so don't do that.
2. Calculating the CDF P(X <= 99) = I_0.96(1901,100), the regularized gamma function, may be above the level you're supposed to know about here and there's not really anything to write down to show your work, some calculator would just be approximating this function to the specified precision.
3. Chances are, if this is a paper assignment and it's an undergrad class, they want you to use the normal approximation to the binomial distribution and look up the probability on a table in the back of your book. It's stupid to look it up in a table and not use a calculator, but it's par for the course with how this stuff is commonly taught.
There are just a ton of ways to accomplish this, so probably whichever is nearest to what you've been taught in class is where you'd want to go.
This post was edited by Amaston on Oct 5 2015 08:46pm