Balls into Bins problem

Question

So I am having a problem trying to solve the following question.

Lets say with have $x\times n$ balls, where $x$ is a values in range $0 < x \leq 1$ and we have $n$ bins.

I'm trying to figure out what the expected number of bins containing at least $1$ ball in them, but the $x$ is throwing me off, does anyone know how to tackle this?

$x$ will be a value such that $x\times n$ results in an integer.

@coffeemath I believe that n × x must be another whole number otherwise the values don't work out. So, if you had 3 balls x could be 1/2, 2/3, or 1 — user3483844, Feb 27 '17 at 12:22

Simon · Answer 1 · 2017-02-27T12:53:24.063

0

The probability of one certain bin to be not hit by one certain ball is $$1-\frac{1}{n}.$$

There are $nx$ balls in total, so the probability one one bin to be not hit by any ball is $$\left(1-\frac{1}{n}\right)^{xn}$$

The probability of a certain bin to be non-empty is simply the inverse of this, i.e. $$1-\left(1-\frac{1}{n}\right)^{xn}.$$

The expected number of non-empty bins is then simply the sum of this expression over all bins. I.e. the answer you are looking for is $$n-n\left(1-\frac{1}{n}\right)^{xn}$$

for large $n$ you might want to approximate this with an exponential function.

edited Feb 27 '17 at 12:53

answered Feb 27 '17 at 12:32

Simon

5,061

May I ask how the x gets into the exponent place? – user3483844 Feb 27 '17 at 12:48
the exponent $xn$ is the number of balls. I edited the answer to make it clearer. Hope this helps. – Simon Feb 27 '17 at 12:54

Balls into Bins problem

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