Solved – formula for a general form of the coupon collector problem

I stumbled across the coupon collectors problem and was trying to work out a formula for a generalization.

If there are $N$ distinct objects and you want to collect at least $k$ copies of each of any $m$ of them (where $m le N$), what is the expectation of how many random objects you should buy?. The normal coupon collector problem has $m = N$ and $k = 1$.

There are 12 different LEGO figures in a collection. I want to collect 3 copies of each of 10 (any 10) figures. I can buy them randomly one at a time. How many should I expect to buy before I have 3 copies of each of 10 of them?

This is not easy to compute, but it can be done, provided $binom{m+k}{k}$ is not too large. (This number counts the possible states you need to track while collecting coupons.)

Let's begin with a simulation to get some sense of the answer. Here, I collected LEGO figures one million times. The black line in this plot tracks the frequencies of the numbers of purchases needed to collect at least three of ten different figures.


The gray band is an approximate two-sided 95% confidence interval for each count. Underneath it all is a red curve: this is the true value.

To obtain the true values, consider the state of affairs while you are collecting figures, of which there are $n=12$ possible types and you wish to collect at least $k=3$ of $m=10$ different types. The only information you need to keep track of is how many figures you haven't seen, how many you have seen just once, how many you have seen twice, and how many you have seen three or more times. We can represent this conveniently as a monomial $x_0^{i_0} x_1^{i_1} x_2^{i_2} x_3^{i_3}$ where the $i_j$ are the associated counts, indexes from $k=0$ through $k=t$. In general, we would use monomials of the form $prod_{j=0}^k x_j^{i_j}$.

Upon collecting a new random object, it will be one of the $i_0$ unseen objects with probability $i_0/n$, one of the objects seen just once with probability $i_1/n$, and so forth. The result can be expressed as a linear combination of monomials,

$$x_0^{i_0} x_1^{i_1} x_2^{i_2} x_3^{i_3}to frac{1}{n}left(i_0 x_0^{i_0-1}x_1^{i_1+1}x_2^{i_2}x_3^{i_3} + cdots + i_3 x_0^{i_0}x_1^{i_1}x_2^{i_2-1}x_3^{i_3}right).$$

This is the result of applying the linear differential operator $(x_1 D_{x_0} + x_2 D_{x_1} + x_3 D_{x_2} + x_3 D_{x_3})/n$ to the monomial. Evidently, repeated applications to the initial state $x_0^{12}=x_0^n$ will give a polynomial $p$, having at most $binom{n+k}{k}$ terms, where the coefficient of $prod_{j=0}^k x_j^{i_j}$ is the chance of being in the state indicated by its exponents. We merely need to focus on terms in $p$ with $i_3 ge t$: the sum of their coefficients will be the chance of having finished the coupon collecting. The whole calculation therefore requires up to $(m+1)binom{n+k}{k}$ easy calculations at each step, repeated as many times as necessary to be almost certain of succeeding with the collection.

Expressing the process in this fashion makes it possible to exploit efficiencies of computer algebra systems. Here, for instance, is a general Mathematica solution to compute the chances up to $6nk=216$ draws. That omits some possibilities, but their total chances are less than $10^{-17}$, giving us a nearly complete picture of the distribution.

n = 12; threshold = 10; k = 3;  (* Draw one object randomly from an urn with `n` of them *) draw[p_] :=    Expand[Sum[Subscript[x, i] D[#, Subscript[x, i - 1]], {i, 1, k}] +        Subscript[x, k] D[#, Subscript[x, k]] & @ p];  (* Find the chance that we have collected at least `k` each of `threshold` objects *) f[p_] := Sum[   Coefficient[p, Subscript[x, k]^t] /.     Table[Subscript[x, i] -> 1, {i, 0, k - 1}], {t, threshold, n}]  (* Compute the chances for a long series of draws *) q = f /@ NestList[draw[#]/n &, Subscript[x, 0]^n, 6 n k]; 

The result, which takes about two seconds to compute (faster than the simulation!) is an array of probabilities indexed by the number of draws. Here is a plot of its differences, which are the probabilities of ending your purchases as a function of the count:

Figure 2

These are precisely the numbers used to draw the red background curve in the first figure. (A chi-squared test indicates the simulation is not significantly different from this computation.)

We may estimate the expected number of draws by summing $1-q$; the result should be good to 14-15 decimal places. I obtain $50.7619549386733$ (which is correct in every digit, as determined by a longer calculation).

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