So we want to achieve $8$ successes where each trial has a probability of succeeding of $0.3$,
$$P(S) = 0.3.$$
We have $8$ "free" trials, and assuming we don't get lucky and get $8$ successes from our free trials, we can pay $1.00$ dollar for a $10\%$ chance to regain one trial. So say we get $3$ successes from the free trials, then we would need to pay at least $5$ dollars [but expected value of $50$ dollars because $50\times0.1 = 5$ successful $10\%$ chances] to regain the $5$ unsuccessful attempts. What is our expected cost to obtain $8$ successes?
In my attempt I start with the expected number of successes from the $8$ free trials ($n \times p = 8 \times 0.3 = 2.4$) and then start a cycle; we have $5,6$ trials to be regained that we missed on the first trials, which would take an expected $\$56.00$ (or fifty-six $10\%$ chances) to regain.
Then from that we get an expected ($5.6 \times 0.3 =$) $1.68$ successes and have $4.08$ successes total. Wash and repeat until we hit $8$ successes. I'm getting around $\$185.00$ as an answer but I'm not sure of my answer and also curious if there's a better or less tedious way of doing this problem.
Any thoughts are appreciated. Thanks!