It's been awhile since I was an expert in this but I can't figure this out.
Let's say I have an experiment where I draw a marble from a bag, note its color, then replace it, 5 times (5 total picks). The bag has 1 white and 4 black marbles. (the chance of picking a white marble each time is 1/5)
Let's say I run this experiment a bazillion times.
What % of outcomes will have 2 white marbles (only)?
Combinations of 2 of 5 gives 10 (out of 32 total) (31%).
I can't get the math to work out no matter what I do. For the 2 picks, I'd initially say the result would be (10/32)*(1/5)^2*(4/5)^3 but that's an impossibly small number that doesn't add up when you do all combinations
If this could be parameterized that would be great (inputs # of draws and the % of white marbles)