Generalization of Coupon Collector's Problem

Question

Suppose I have a bag containing three different marbles: red, green, and blue. I am drawing a single marble from the bag each time with replacement. I would like to know how many times, on average, do I need to draw marbles from the bag until I have drawn $25$ of each type of marble.

After some research I think this is a multinomial distribution. I can calculate the probability that I have exactly $25$ of each marble after $75$ draws by the following:

$$\frac{75!}{25!\times25!\times25!}\times\left(\frac13\right)^{25}\times \left(\frac13\right)^{25} \times\left(\frac13\right)^{25}$$

which works out to about $1.06\%$. However, I am not confident in this answer because it doesn't match up with my observations in the real world. Any help would be appreciated.

Answer 1

Based on your example, I am assuming that you have equal numbers of each marble.

Let $R,G,B$ be random variables representing the number of red, green, or blue marbles respectively that have been drawn. It sounds like you are looking for

$$P(R=r,G=g,B=b) = \dfrac{(r+g+b)!}{r!g!b!}\left(\dfrac{1}{3}\right)^{r+g+b}$$

Now, you can plot this as you vary $r,g,b$.

As for how many draws do you need on average to draw 25 of each, this is a modified coupon collector's problem. By linearity of expectation, drawing 25 of a single color marble will require $\dfrac{25}{\tfrac{1}{3}} = 75$ pulls of a marble. And drawing 25 of each would require $75+75+75 = 225$ draws on average.

Edit: this coupon collector solution does not work, unfortunately. The chances of getting multiples of the same thing are different from getting one of each of different things. I should have realized that. Through simulation, I am getting much smaller numbers. Like around $87.7$.

This gives a much closer estimate than my ill-conceived use of the Coupon Collector's Problem:

$$3\sum_{a=25}^{75}\sum_{b=25}^{75}(25+a+b)\dfrac{(24+a+b)!}{24!a!b!}\cdot \dfrac{1}{3^{25+a+b}} \approx 87.9057$$

Now, technically, I am undercounting, as $a,b$ should go to infinity, but I am also overcounting by multiplying by 3, as I am counting when $a=b=25$ three times, even though it should only be counted once. So, this is an estimate of the expected value.

Answer 2

Let $E(r,g,b)$ be the expected number of draws to obtain at least $r$ red, $b$ blue, and $g$ green marbles. Then $E(0,0,0)=0$ and otherwise $$E(r,g,b)= 1+\frac{1}{3}\left[E(\max(r-1,0),g,b)+E(r,\max(g-1,0),b)+E(r,g,\max(b-1,0))\right].$$ Solving this recursion yields $E(25,25,25)\approx 87.9057937186312$, in close agreement with @InterstellarProbe's second estimate. A plot of $E(n,n,n)$ for $n=1$ to $50$ is nearly linear:

The Wikipedia article for the Coupon collector's problem gives a reference to Newman and Shepp (1960) for this generalization, and the exact value is $$E(25,25,25)=3 \int_{0}^\infty \left[1-\left(1-e^{-t}\sum_{k=0}^{25-1} \frac{t^k}{k!}\right)^3\right] \mathrm{d}t.$$