Investigating the Statistics of the Number of Dice-Rolls Until Reaching a Desired Goal

Lucy MARTINEZ and Doron ZEILBERGER

Everybody knows that the dice are loaded - Leonard Cohen

Abstract: We harness the great power of symbolic computation and experimental mathematics to investigate the statistics of the number of (loaded) dice-rolls it takes until every face showed at least a certain desired number of times. We also investigate much more general scenarios.

Preface:

In a recent fascinating article that appeared in the Mathematical Intelligencer, Simon Blatt, Uta Freiberg, and Vladimir Shikhman investigated the mathematics of Talmudic Family Planning, where a couple has a predecided goal to “Go forth and multiply” until they have begotten n boys and k girls, for some pre-decided n and k. They stop as soon as they have reached both goals. They assumed that the probability of having a boy, p, is always the same, and the gender of each birth is independent of the others. They were wondering about the expected family size, that they called F (n, k, p) in terms of n, k, and p, and proved the elegant constant gender rule, that asserts that if B(n, k, p) and G(n, k, p) are the expected number of boys and expected number of girls, respectively (of course F (n, k, p) = B(n, k, p) + G(n, k, p)) then

\frac{B(n, k, p)}{G(n, k, p)} = \frac{p}{1 = p} .

Of course, mathematically this is equivalemt to tossing a coin whose probability of Heads is p (and hence the probability of Tails is (1 − p)) and stopping as soon as you have encountered at least n Heads and at least k Tails. The minimum number of coin-tosses is n + k, but the total number is arbitrary.

In this paper we will generalize it to an arbitrary die, with k faces, let’s call them 1, 2, …, k, where the probability of landing on face i is p_{i}, and of course

∑_{i=1}^{k} p_{i} = 1 .

References

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Lucy Martinez and Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: lm1154 at math dot rutgers dot edu , DoronZeil at gmail dot com .

Written: Oct. 15, 2025.