I draw with replacement from a pot containing 10 balls: 7 red, 3 white. If I draw a red ball I score 2 points, and if I draw a white ball I score 1 point. I stop drawing when I reach 10 or more points.
I am looking to find the distribution of $n$, the number of draws I make before stopping.
I can use brute force to define and sample from this distribution without issue. But I am having trouble defining a closed form mathematical solution.
I'm guessing the distribution will be related to the binomial distribution since it is discrete and we are sampling with replacement. However although we have a defined trial success probability (e.g 7/10 = 0.7), I'm looking to find the number of trials, $n$, which is variable and not the number of successes (although this value would be related to the number of trials needed to reach 10)
Is there a family of discrete distributions that fits this type of problem or would this require some kind of compound distribution?