I plan to host a lucky draw where every player who enters the lucky draw wins a 2 dollar prize with probability 0.8, wins a 5 dollar prize with probability 0.15, wins a 10 dollar prize with probability 0.049, and wins the 1000 dollar prize with probability 0.001.
I plan to limit the first 500 players to my draw. Once someone wins the 1000 dollar prize, the lucky draw ends immediately. If none of the 500 players wins the 1000 dollar prize, then the lucky draw ends.
How do I find the expected total dollar value of prizes given out for the lucky draw?
I wrote a Python simulation code that yields an estimated mean of 1515 dollars with a standard deviation of 282 dollars. Is there a more rigorous way to solve this using probability theory?
The expected value of a draw, contingent on it not being the big payout, is about $2.8428$. The possible paths are either $500$ of those draws or of the form $X^iY$ where $i\in \{0,1,\cdots, 499\}$. Here $X$ denotes any draw other than the big one, and $Y$ denotes the big one. The probability of a non-big draw is $.999$, so $$P(X^iY)=.999^i\times .001$$ and of course $P(X^{500})=.999^{500}$ The expected payout from $X^{500}$ is $2.8428\times 500$ and the expected payout from $X^iY$ is $2.8428\times i+1000$.
Thus the answer is $$500\times 2.8428+2.8428\times .001\times \sum_{i=0}^{499}i \times .999^i+1000*(1-.999^i)=\boxed {1511.5}$$
confirming your result (at least roughly).
The sum can be computed via Geometric Series (or rather, by a standard modification of Geometric Series), though I just did it via Wolfram Alpha