<s>[INST] Rolling a fair die You and your friend play a game in which you and your friend take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. Play continues until either player wins if, after the player rolls, the number
on the running tally is a multiple of 7.  Should you start first or it is better if you let your friend rolls the die first.
 [/INST] I thought I'd answer this question as another opportunity to learn some math that's way out of my league in hopes that some of my usual favorites will evaluate it and see if I'm on the right track.
The first thing I did was consider a few trials:


*

*Player 1 has a 0/6 chance of winning

*Player 2 has a 1/6 chance of winning

*Player 1 has a 1/6 chance of winning

*Player 2 has a 1/6 chance of winning


The probability of player 1 winning a 1 round game is 0.  The probability of player 1 winning a 2 round game is $\mathbb{P} = \mathbb{P}(\text{Winning Round 1}) + \mathbb{P}(\text{Rounds 1 and 2 Losses}) \cdot \mathbb{P}(\text{Winning Round 3}))$.
I figured that continues infinitely with the outcomes becoming less and less likely as the number of trials approaches infinity.
So, I came up with this:
$$\mathbb{P}(\text{Player 1}) = \dfrac{0}{6} + \dfrac{6}{6} \cdot \dfrac{5}{6} \cdot \dfrac{1}{6} + \dfrac{6}{6} \cdot \left(\dfrac{5}{6}\right)^3 \cdot \dfrac{1}{6} + \ldots $$
$$\mathbb{P}(\text{Player 1}) = 0 + \dfrac{5}{6} \cdot \dfrac{1}{6} + \left(\dfrac{5}{6}\right)^3 \cdot \dfrac{1}{6} + \ldots = \dfrac{1}{6} \cdot \left( \dfrac{5}{6} + \left(\dfrac{5}{6}\right)^3 + \left(\dfrac{5}{6}\right)^5 + \ldots \right) $$
So, then I had to figure out how to sum an infinite series: $\left( \dfrac{5}{6} + \left(\dfrac{5}{6}\right)^3 + \left(\dfrac{5}{6}\right)^5 + \ldots \right) $
I did some research on how exactly that's done and the best I could come up with is to multiply the entire series by $\left(\dfrac{5}{6}\right)^2$ and subtract that from the original series.  So, I think this is valid:
$$\left(\dfrac{5}{6}\right)^2 \cdot \left( \dfrac{5}{6} + \left(\dfrac{5}{6}\right)^3 + \left(\dfrac{5}{6}\right)^5 + \ldots \right) = \left( \left(\dfrac{5}{6}\right)^3 + \left(\dfrac{5}{6}\right)^5 + \ldots \right)$$
If I subtract the new series from the original, all of the elements but the first should cancel out and I should have this:
$$\mathbb{S} - \left(\dfrac{5}{6}\right)^2 \cdot \mathbb{S} = \dfrac{5}{6}$$
$$\mathbb{S} \cdot \left[1 - \left(\dfrac{5}{6}\right)^2\right] = \dfrac{5}{6}$$
$$\mathbb{S} = \dfrac{\dfrac{5}{6}}{\left[1 - \left(\dfrac{5}{6}\right)^2\right]}$$
If I plug that into my original formula, I get this:
$$\mathbb{P}(\text{Player 1}) = \dfrac{1}{6} \cdot \dfrac{\dfrac{5}{6}}{\left[1 - \left(\dfrac{5}{6}\right)^2\right]} = \overline{.45}$$
Because there are only two players this should be true: $\mathbb{P}(\text{Player 2}) = 1 - \mathbb{P}(\text{Player 1}) = \overline{.54} $
Unsure of my original conclusion, I thought I'd calculate the probability of Player 2 winning to verify.
$$\mathbb{P}(\text{Player 2}) = \dfrac{6}{6} \cdot \dfrac{1}{6} + \dfrac{6}{6} \cdot \left(\dfrac{5}{6}\right)^2 \cdot \dfrac{1}{6} + \dfrac{6}{6} \cdot \left(\dfrac{5}{6}\right)^4 \cdot \dfrac{1}{6} + \ldots $$
$$\mathbb{P}(\text{Player 2}) = \dfrac{1}{6} + \dfrac{1}{6} \cdot \left(\left(\dfrac{5}{6}\right)^2 \cdot \left(\dfrac{5}{6}\right)^4 + \left(\dfrac{5}{6}\right)^6 + \ldots \right) $$
To sum that series I thought: $\mathbb{S} = \dfrac{25}{36} + \left(\dfrac{25}{36}\right)^2 + \left(\dfrac{25}{36}\right)^3 + \ldots $
So, $\mathbb{S} - \dfrac{25}{36}\mathbb{S} = \dfrac{25}{36}$ and $\mathbb{S} = \dfrac{\dfrac{25}{36}}{1 - \dfrac{25}{36}}$
I plug that into the original formula and find that $\mathbb{P}(\text{Player 2}) = \overline{.54}$ which is what I expected from before.
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