A die is thrown until a $6$ appears. If the casting stops on an odd number of times Mike wins. Otherwise, Tom wins. What is the probability that Mike wins? What is the probability that Tom wins? Also show that this game favors mike for all $p$.
Attempt: $\Pr(\text{Mike wins}) = \sum_{k=1}^\infty \left(\frac 5 6\right)^{k-1} \frac 5 6\text{ for }k = 1,3,5,\ldots,\infty$
$\Pr(\text{Tom wins}) = \sum_{k=1}^\infty \left(\frac 5 6\right)^{k-1}\frac 5 6\text{ for }k = 2,4,6,\ldots, \infty$
I am stuck here.