I recently ran into this problem on an assingment. Supose you have a fair dice and let $X$ be the random variable that counts how many tosses happen before the sum of the numbers that showed up in each toss adds up to something greater than $6$.
You're asked to find the CDF of $X$ which has support in $\{2,\dots,7\}$, but I can't figure out a closed form for it. I've tried calculating the probability function $\mathbb{P}[X=k]$ for each $k$ in the support of $X$ but it's a bit messy. Any ideas?
The key is working with the definition of CDF. We want to find the probability that the sum is greater than 6 in at $\textit{most}$ $x$ rolls. Now, consider $1 - P(X \leq x)$. This is the probability that the sum is greater than 6 in at $\textit{least}$ $x+1$ rolls. But this is equivalent to the probability that the sum is less than or equal to 6 in $\textit{exactly } x$ rolls. To calculate this new probability, we can sum across possible roll sums. The number of ways in which $k$ rolls can sum to $n \le 6$ = ${n-1 \choose k-1}$, so: $\displaystyle{P(X \leq x) = 1 - (1 - P(X \leq x) = 1 - \sum_{i=x}^{6}{\frac{{i-1 \choose x-1}}{6^x}} = 1 - \frac{6 \choose x}{6^x}}$.
Note: if $x \geq 7$, then ${6 \choose x} = 0$, so the CDF becomes $1 - \frac{0}{6^x} = 1$.