I can't seem to crack this nut and it would be great if someone has a hint/solution. I think a clever application of the Central Limit theorem is needed here, but I don't know how.
Suppose I roll a 6-sided die $N$ times. However I stop rolling as soon as the total sum of the results exceeds 700, so I have $117\leq N\leq 700$ almost surely (extreme cases of only 6's and 1's used as bounds).
I would now like to estimate the probability, that I rolled the die say more than 210 times. Can someone help me out?
Instead of considering the more complicated problem where the number of dice rolls $N$ is random, suppose that we simply roll the dice $210$ times. Let $S = \sum_{i=1}^{210} X_i$ represent the total sum of the results, where each $X_i$ is the result of each independent dice roll. Note that if $S < 700$, in the original problem you would have to keep rolling (i.e. $N > 210$). However, if $S \geq 700$, then we would have stopped at or before all 210 rolls finished (i.e. $N \leq 210$). Therefore, the event you are interested in is simply the probability $$\mathbb{P}(S < 700) = \mathbb{P}\left( \sum_{i=1}^{210} X_i < 700 \right).$$ Can you see how the CLT may be applied now?