<s>[INST] Probability Dice Game Paul, Dave and Sarah are rolling a fair six sided die. Paul will go first, always followed by Dave, who is always followed by Sarah, who is always followed by Paul, and so on...
What is the probability that Sarah will be the first one to roll a six?
Now I know the probability of Sarah rolling a six first on the first sequence would be
(5/6)(5/6)(1/6)
But what would be the probability of her rolling a six, if she failed to on the first sequence, the second sequence.
This lead to me finding the equation
(((5/6)(5/6)(5/6))^s)((5/6)(5/6)(1/6))
The first part being the number of sequences, s, she did not roll a six and the second part being when she finally rolls a six.
So I am doing this right?
Could someone help me figure out what to do next, or if I am completely wrong point me in the right direction?
 [/INST] The probability that Sarah is the first to roll a $6$ is $\frac{25}{91}$. Think it this way: Suppose $n$ people roll a die, one by one, in a fixed order, until someone gets a $6$. Let $P_{ij}$ be the probability that player $i$ gets that first $6$ at round $j$, we have:
· Round $1$:
$P_{11} = \frac{1}{6}, \, P_{21} = \frac{1}{6}\left(\frac{5}{6}\right), \, P_{31} = \frac{1}{6}\left(\frac{5}{6}\right)^{2}, \, \ldots \,, \, P_{n1} = \frac{1}{6}{\left(\frac{5}{6}\right)}^{n-1}$
· Round $2$:
$P_{12} = {\left(\frac{5}{6}\right)}^{n}\frac{1}{6}, \, P_{22} = {\left(\frac{5}{6}\right)}^{n+1}\frac{1}{6}, \, P_{32} = {\left(\frac{5}{6}\right)}^{n+2}\frac{1}{6}, \, \ldots \,, \, P_{n2} = {\left(\frac{5}{6}\right)}^{2n-1}\frac{1}{6}$
· Round $3$:
$P_{13} = {\left(\frac{5}{6}\right)}^{2n}\frac{1}{6}, \, P_{23} = {\left(\frac{5}{6}\right)}^{2n+1}\frac{1}{6}, \, P_{33} = {\left(\frac{5}{6}\right)}^{2n+2}\frac{1}{6}, \, \ldots \,, \, P_{n3} = {\left(\frac{5}{6}\right)}^{3n-1}\frac{1}{6}$
· Round $k$:
$P_{1k} = {\left(\frac{5}{6}\right)}^{(k-1)n}\frac{1}{6}, \, P_{2k} = {\left(\frac{5}{6}\right)}^{(k-1)n+1}\frac{1}{6}, \, P_{3k} = {\left(\frac{5}{6}\right)}^{(k-1)n+2}\frac{1}{6}, \, \ldots \,, \, P_{nk} = {\left(\frac{5}{6}\right)}^{kn-1}\frac{1}{6}$
So $P_{ij} = \frac{1}{6}{\left(\frac{5}{6}\right)}^{(j-1)n + (i-1)}$. The probability that player $i$ gets the first $6$ is the sum of the probabilities over each of his turns: $P_i = \displaystyle \sum_{j = 1}^{\infty} \frac{1}{6}{\left(\frac{5}{6}\right)}^{(j-1)n + (i-1)} = \frac{{\left(\frac{5}{6}\right)}^{i-1}}{6\left(1 - {\left(\frac{5}{6}\right)}^{n}\right)}$
In your particular case, $n=3$ and Sarah is the third player so $i=3$.
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