We flip a fair coin until we obtain our first tails. Given the first tails occurs on the nth flip, we roll a fair 6−sided die n times. Find the probability that the die value 1 is observed in the rolls.
Probability 1 is seen given n will be $1 - \frac56^n = \frac{6^n-5^n}{6^n}$. The probability the number of coin flips is n is $\frac12^n$. So we can say the probability 1 is observed is the product of these probabilities (as they will result in the joint probability of seeing 1 for that value of n) summed for all values n resulting in: $\sum_{n=1}^\infty\frac{6^n-5^n}{12^n}$. However, I am not exactly sure how to solve this and am thinking there is probably an easier/more intuitive route to go about this problem.