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As seen here. Assume that each cascader bubble begins with a d6, as opposed to being determined by the prime bubble along with the number of cascader bubbles; the scene makes it ambiguous.

$\frac{1}{36} = 0.02777...$ provides a simple lower bound for the probability (1 on the prime bubble and the sole cascader bubble). If the prime bubble rolls a 2, the odds are $\frac{1}{6}\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{18}+\frac{1}{24}+\frac{1}{30}+\frac{1}{36}\right) = \frac{49}{720} = 0.0680555...$; multiple that by the 1/6 chance of rolling that 2 and add to the original 1/36, and you get $0.02\bar{7}+0.01134\overline{259} = 0.03912\overline{037}$. After that, it gets way beyond me.

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    $\begingroup$ Formally: let $X_0,X_1, X_2, X_3, ...$ be a sequence of random variables. $X_{0}$ is equal to $n$ with probability one (where $n$ is the number of sides on the initial cascading dice). Each successive variable $X_{k+1}$ is uniformly distributed on $\{1,2,3,...,\Pi_{i=0}^{k} X_{i}\}$. Define $p_{k}=P[X_{k}=1]$; what is $(1/6)\sum_{i=1}^{6}p_i$? $\endgroup$
    – mjqxxxx
    Sep 10, 2012 at 14:24
  • $\begingroup$ I found ambiguous whether there were $6$ more dice all the time or the number matched the original roll, not the number of sides on the dice. The chance if the prime bubble rolls higher than $2$ that you get a $1$ at the end will be very small, so in practice you can ignore it. $\endgroup$
    – Ross Millikan
    Oct 6, 2012 at 2:52

1 Answer 1

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Suppose the prime bubble rolls an $n$. Then the cascader bubble rolls will be $\{r_1, r_2, ... r_{n-1},1\}$ with probability $$ \frac{1}{6}\cdot\frac{1}{6r_1}\cdot\frac{1}{6r_1r_2}\cdots\frac{1}{6r_1r_2\cdots r_{n-1}}=\frac{1}{6^n r_1^{n-1} r_2^{n-2} \cdots r_{n-1}} $$ for any $r_1\in[6]=\{1,2,3,4,5,6\}$, $r_2\in[6r_1]$, $r_3\in[6r_1r_2]$, etc., up to $r_{n-1}\in[6r_1r_2\cdots r_{n-2}]$. The total probability is then $$ p_{n}=\frac{1}{6^n}\sum_{r_1=1}^{6}\frac{1}{r_1^{n-1}}\sum_{r_2=1}^{6r_1}\frac{1}{r_2^{n-2}}\cdots\sum_{r_{n-1}=1}^{6r_1r_2\cdots r_{n-2}}\frac{1}{r_{n-1}}. $$ In particular, $$ \begin{eqnarray} p_1&=&\frac{1}{6}=0.166666... \\ p_2&=&\frac{1}{36}\sum_{r_1=1}^{6}\frac{1}{r_1}=\frac{1}{36}H_{6}=\frac{49}{720}=0.06805555... \\ p_3&=&\frac{1}{216}\sum_{r_1=1}^{6}\frac{1}{r_1^2}\sum_{r_2=1}^{6r_1}\frac{1}{r_2} \\ &=&\frac{1}{216}\left(H_{6}+\frac{1}{4}H_{12}+\frac{1}{9}H_{18}+\frac{1}{16}H_{24}+\frac{1}{25}H_{30}+\frac{1}{36}H_{36}\right) \\ &=&\frac{97493779762855253}{5104009215002880000}=0.01910141... \end{eqnarray} $$ Using these first three terms gives the lower bound $(p_1+p_2+p_3)/6 = 0.04230...$.

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