<s>[INST] Middle School Probability Problem 
Sven and Ole take turns rolling a standard six-sided die. The first person to roll
  a six wins. If Sven goes first, what is the probability that he will win the game?

This is a problem from a worksheet for middle schoolers.  I see that one could solve it by summing the infinite series
$$P(\text{Sven wins}) = P(\text{Sven wins in 1st round}) + P(\text{Sven wins in 2nd round}) + \cdots$$ but that's surely not how it is supposed to be solved.  How else could one solve this?
 [/INST] Let $A$ be the event in which Sven roll $6$ and $B$ be the event in which Ole rool $6$
So $\displaystyle P(A)=\frac{1}{6}$ and $\displaystyle P(\bar{A}) = \frac{5}{6}$ and $\displaystyle P(B)=\frac{1}{6}$ and $\displaystyle P(\bar{B}) = \frac{5}{6}$ 
So Required Probability 
$$\displaystyle = P(A)+P(\bar{A})\cdot P(\bar{B})\cdot P(A)+P(\bar{A})\cdot P(\bar{B})\cdot P(\bar{A)}\cdot p(\bar{B})\cdot P(A)+.......\infty$$
So required probability $$\displaystyle = \frac{1}{6}+\frac{5}{6}\cdot \frac{5}{6}\cdot \frac{1}{6}+\frac{5}{6}\cdot \frac{5}{6}\cdot \frac{5}{6}\cdot \frac{5}{6}\cdot \frac{1}{6}+.....\infty$$
So Required Probability $$\displaystyle = \frac{1}{6}\left[\frac{1}{1-\left(\frac{5}{6}\right)^2}\right] = \frac{6}{11}.$$
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