Knowing I did mathematics my room mate posed me a question yesterday, he wanted to know given a number of tracks randomly played what is the average of a track being first repeated. He'd already written a program in Perl to solve this problem by brute calculation but was coming up with some very odd answers, he later discovered the random function wasn't very good.(adsbygoogle = window.adsbygoogle || []).push({});

Now, I was some what rusty on my probability skills and had to take this away to think about it. This did and still does sound like a very doable problem. This is how I started out:

Let X be the track number where the first repeat is, where the initial track played is track 0. Let x_{1}be the event where X=1, x_{2}be the event where X=2 etc.. To start off easy, let there be 3 tracks, then:

[tex]P(x_1) = \frac{1}{3}[/tex]

It follows then that:

[tex]P(x_2 | {x_1}') = \frac{2}{3}[/tex]

(In words, the probability of x_{2}occurring given that x_{1}has not occurred is 2/3). It therefore follows from some basic rules of probability:

[tex]\frac{P({x_1}' \cap x_2)}{P({x_1}')} = \frac{2}{3}[/tex]

I then drew a little diagram to show that x_{2}was a subset of x_{1}'. And therefore:

[tex]P(x_2) = \frac{4}{9}[/tex]

I then worked out the probability of x_{3}the long way to make sure I was doing everything right, by saying that:

[tex]P(x_3|({x_2}' \cap {x_1}')) = 1[/tex]

After expanding and reducing this nicely led to:

[tex]P(x_3) = \frac{2}{9}[/tex].

This gave a nice expectation of x as:

[tex]E(x) = \frac{17}{9}[/tex]

I then set out looking at n tracks, which was pretty much the same:

[tex]P(x_1) = \frac{1}{n}[/tex]

[tex]P(x_2 | {x_1}') = \frac{2}{n}[/tex]

Which led to:

[tex]P(x_2) = \frac{2}{n} - \frac{2}{n^2}[/tex]

I also worked out:

[tex]P(x_i | ( {x_{i-1}}' \cap \ldots \cap {x_1}')) = \frac{P(x_i)}{P({x_{i-1}}' \cap \ldots \cap {x_1}')}[/tex]

Which leads to:

[tex]P(x_i) = \frac{i}{n} \left(1 - \sum_{r=1}^{i-1} (P(x_r)) \right) \quad \text{with} \quad P(x_1) = \frac{1}{n}[/tex]

This is far too complicated for me to deal with, so I attempted to put in to mathematica and it doesn't seem to be able to make anymore out of it that I can either. I feel I am going wrong somewhere because this doesn't seem like too difficult of a problem.

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# Homework Help: Average Number of tracks

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