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A friend of mine inspired this question:

Given a fair die, can one accurately estimate (within some margin) the size of the die given enough rolls? For example: If I roll a die 1000 times and all the numbers fall on and between 1 and 5, I estimate the size of the die is 5.

My question is: is this an accurate estimate? Will we always be able to predict the size of the die within some margin of error?

Any help is appreciated.

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  • $\begingroup$ Are you assuming that you know that it starts with a 1 and goes up (without skipping any numbers)? $\endgroup$ – TravisJ Mar 14 '15 at 21:13
  • $\begingroup$ Do you know how many sides the die has? $\endgroup$ – John Mar 14 '15 at 21:15
  • $\begingroup$ To clarify: We're assuming the die starts at 1 and goes up to some number n. The only information we are given is the results of k rolls. The question then becomes: How large does k have to be in order to get a good enough estimate? $\endgroup$ – Rambo V Mar 14 '15 at 21:28
  • $\begingroup$ I'm still unclear as to whether you know how many sides the die has. Does "goes up to some number $n$" mean it has n sides numbered $1$ to $n$, or that it has some known number of sides with a maximum value of $n$ on one (or more) of the sides? $\endgroup$ – John Mar 14 '15 at 22:11
  • $\begingroup$ Like the post says: the only information you are given is the result of k rolls. This does NOT include the size of the die. $\endgroup$ – Rambo V Mar 14 '15 at 23:49
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As Bruce has said in the comments, the most obvious choice for the estimator of the number of sides is the maximum of all the $k$ rolls seen. Let's take that and see what we can learn about it.

Let $x_i$ be the result of the $i$th roll. Then our estimator is $$ \hat{N} \equiv \max_i \; x_i. $$ So the way that the estimator would fail, would be if the largest value of the die doesn't come up in the $k$ rolls. Because the die is assumed to be fair the probability distribution for each value is equal: $$ P(x_i = n) = \frac{1}{N}, \qquad \forall \quad i, \text{and for any } n. $$ The probability that $N$ doesn't get rolled would be $$ P(x_i \ne N) = 1 - P(x_i = N) = 1 - \frac{1}{N}. $$ Over the full course of $k$ rolls, we can calculate the probability of never rolling the largest value $(N)$ as $$ P\left(\bigcap_{i=1}^k (x_i \ne N)\right) = \prod_{i=1}^k P(x_i \ne N) = \left( 1- \frac{1}{N}\right)^k = \left(\frac{N-1}{N}\right)^k. $$ Now say that we want to compare this with some margin of error $\epsilon$. ( For example, an $\epsilon= 0.01$ would represent a 1% possibility that the largest number doesn't show up in our $k$ rolls.) $$ \begin{split} \left(1-\frac{1}{N}\right)^k &\le \epsilon \\ k\;\underbrace{\log\left(1 - \frac{1}{N}\right)}_{<\,0} &\le \log(\epsilon) \\ k &\ge \frac{\log{\epsilon}}{\log\left(1-\frac{1}{N}\right)} \end{split} $$ So, given a margin $\epsilon$ set to any arbitrary value (well, it must be positive and less than 1), and the number of sides of the die, we can tell how many rolls $(k)$ we need to achieve that confidence. A couple of tables below show some values of $k$ given an $N$. $$ \epsilon = 0.01 $$ $$ \begin{array}{l|cccccccccc} N & 2 & 4 & 6& 8& 10 &12 &20 &26 &40 &100\\ \hline k & 7 & 17 &26 &35 &44 &53 &90 &118 &182 &459 \end{array} $$ $$ \epsilon = 0.001 $$ $$ \begin{array}{l|cccccccccc} N & 2 & 4 &6 &8 &10 &12 &20 &26 &40 &100 \\ \hline k& 10 &25 &38 &52 &66 &80 &135 &177 &273 &688 \end{array} $$ Alternatively, we can plug in the number of rolls and see how large a die we can identify with a set margin for error. This can be found by backing out what $N$ can be identified within a margin $\epsilon$ with a given $k$. Working it backwards from above, you get something that looks like $$ N \le \frac{1}{1 - e^{\frac{\log(\epsilon)}{k}}}, $$ which says that for $k =1000$, you'd be able to identify a 216-sided die with a 99% probability, and an 87-sided die with a 99.999% probability.

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