# How would I calculate a combination of the Binomial and Geometric Distributions?

To be specific with my problem, I'm calculating a formula for a game. There's 6 independent trials, each with an independent probability of success = 0.34. I know that from the Geometric distribution, it'll take about 3 trials before I get a success, so should expect about 2 successes for every 6 trials?

However, my issue is that anytime I get a success, I get another trial, so if I get 2 successes in 6 trials, then I can get 2 more trials, and I'm having trouble determining the amount of successes I can expect on average, although I have a feeling it's around 3.

The appropriate distribution for the number of successes before some fixed number of failures from i.i.d. Bernoulli trials is the negative binomial distribution. The shape parameter $$r$$ would in this case be equal to six and the probability parameter $$p = 0.34$$ (in the parameterization on the Wikipedia page). The expected value of the number of successes is $$rp/(1-p)$$, which equals 3.09, pretty close to your intuition!