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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!

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Yes there is a simpler way. $\frac{8}{0.3} = 26.667$ which rounds up to $27$ trials to get an expected $8$ successes. So you have to pay for another $19$ trials for which the expected number of $\$1$ payments would $\frac{19}{0.1} = 190$ hence the expected cost would be $\$190$.

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    $\begingroup$ If this is an expected value, then there should be no issue I think with the expected number of required trials being fractional. $\endgroup$ – Riley Dec 17 '18 at 3:37

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