What Is the Average Number of Plays Before a Track Repeats?

[Zurtex]

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.

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 x₁ be the event where X=1, x₂ be the event where X=2 etc.. To start off easy, let there be 3 tracks, then:

$$P(x_1) = \frac{1}{3}$$

It follows then that:

$$P(x_2 | {x_1}') = \frac{2}{3}$$

(In words, the probability of x₂ occurring given that x₁ has not occurred is 2/3). It therefore follows from some basic rules of probability:

$$\frac{P({x_1}' \cap x_2)}{P({x_1}')} = \frac{2}{3}$$

I then drew a little diagram to show that x₂ was a subset of x₁'. And therefore:

$$P(x_2) = \frac{4}{9}$$

I then worked out the probability of x₃ the long way to make sure I was doing everything right, by saying that:

$$P(x_3|({x_2}' \cap {x_1}')) = 1$$

After expanding and reducing this nicely led to:

$$P(x_3) = \frac{2}{9}$$.

This gave a nice expectation of x as:

$$E(x) = \frac{17}{9}$$

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

$$P(x_1) = \frac{1}{n}$$

$$P(x_2 | {x_1}') = \frac{2}{n}$$

Which led to:

$$P(x_2) = \frac{2}{n} - \frac{2}{n^2}$$

I also worked out:

$$P(x_i | ( {x_{i-1}}' \cap \ldots \cap {x_1}')) = \frac{P(x_i)}{P({x_{i-1}}' \cap \ldots \cap {x_1}')}$$

Which leads to:

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

This is far too complicated for me to deal with, so I attempted to put into 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.

 

[shmoe]

It looks like

$$P(x_i)=\frac{i}{n}\prod_{j=0}^{i-1}\frac{n-j}{n}=i\frac{n!}{n^{i+1}(n-i)!}$$

offhand I can't see a nice expression for the expected value, but you can find it easily enough for a given n.

 

[Hurkyl]

This sounds like the Birthday Problem.

The general rule of thumb is that out of N choices, it takes on the order of $\sqrt{N}$ picks to get a repeat. (Actually, one ought to multiply that by something to get a better estimate... probably something like $\sqrt{2}$, but I can't remember for sure)


The analysis on the Wikipedia page says the probability of getting no repeats after K picks is about:

$$\bar{P}(K) = e^{-\frac{K^2}{2 N}}$$


You're asking for the expected number of picks before getting a repeat. In other words:

$$\sum_{k = 1}^{N} k \bar{P}(k-1) \frac{k}{N}$$

because the odds that the first repeat happens at pick k is the product of the odds of no repeats by the (k-1)-th pick and the odds that the k-th pick is one of those previously chosen.

Well, I can do another approximation by pretending this is an integral and not a sum, and goes from 0 to infinity:

$$E \approx \int_0^{+\infty} \frac{1}{N} k^2 e^{-\frac{K^2}{2 N}} \, dx$$

if I've done my integration by parts and substitutions correctly, this yields:

$$E \approx \sqrt{2N} \int_0^{+\infty} e^{-x^2} \, dx = \sqrt{\frac{N \pi}{2}}$$

 

[Zurtex]

Cool, that's very nice, I did a little work from what shmoe gave me and got an exact expectation, which is given n tracks then:

$$E(x) = -1 + n e^n \text{ExpIntegralE}[-n,n]$$

Which if you look at definitions and stuff isn't too far out from your approximation.

 

[shmoe]

As n goes to infinity, that's asymptotic to what Hurkyl provided.

That's quite a lot of approximations in Hurkyl's post! the sums to integrals are fine, the integrand is increasing up to sqrt(2N) and decreasing thereafter with a max of 2/e at sqrt(2N), so this will be off by a constant at most.

What seems harder to justify is replacing the exact probability with $$\bar{P}(K)$$, which is a terrible approximation when K is close to N. However, it's an upper bound for the true probability (1+x<e^x) and it drops of very quickly, so the main contribution for the expected value sum will come from something like the first $$N^{1/2+\epsilon}$$ terms. This should handily show that the expected value sum in this range is asymptotic to the approximation.

 

[Hurkyl]

Ack, you don't want me to do some real work, do you? :uhh:

 

[balakrishnan_v]

Let us find the fraction of the number line(from 0 to 1) that will be filled:
Now we will find E(1) which is nothing but the expected fraction of the line(x=0 to x=1) given that each filling width is a small h
(h=1/N)
E(1)=h+E(1-h)
E(1-h)=(1-h){h+E(1-2h)}
and so on
hence we get
E(1)=h{1+(1-h)+(1-h)(1-2h)+.,,}
Now since h is very small we can make (1-kh)->exp(-kh)
So the nth term->(exp(-(n^2+n)/2)->exp(-hn^2/2)
so the sum can be approxuimated to the integral
Note that kh<<1
Hence
$$\int_{0}^{\infty} e^{-\frac{hx^2}{2}}dx=\sqrt{\frac{\pi h}{2}}$$
The above is wrt to 0 to 1
So if we scale it up to N and noting that h=1/N since
$$\sqrt{\frac{\pi N}{2}}$$