Hey guys so the question is you are randomly throwing $100$ balls into $20$ bins, how many bins (on average) will end up empty?.
Assume that n balls are thrown randomly and independently into k bins. for the first example, we used 10 balls and 10 bins. So $N=10$ and $K=10$
The equation I used was $\left(\frac{k-1)}{k}\right)^n$ . Multiply by $k$ for the expected number of empty bins.
Your equation will look like $\left(\frac{10-1}{10}\right)^{10} \cdot 10 = 3.486784401$ which makes sense as an answer for the amount of bins empty.
When plugging my numbers into the equation we get $\left(\frac{20-1}{20}\right)^{100} \cdot 20 = 0.1184105844$.
Does my answer that I am getting make sense or is a number less than one not plausible. It is not a whole number and does not make sense why the average number of bins would be less than one. any help on what I could be doing wrong would be appreciated.
I also used an equation that is similar and produces the same results, it was $m(1-(1/m))^n$
Where $n=$ balls and $m=$ bins.
Thanks Steven
Your calculation is correct. The expected number of empty bins is less than one because it's very unlikely to leave an empty bin when you throw 200 balls into 20 bins. So when you attempt this exercise, most of the time the number of empty bins is zero, and a small fraction of the time it is one or more. Averaging this out gets you a number less than 1.