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**noob mathematician** A casino patron will continue to make $5 bets on red in roulette until she has won 4 of these bets.

On each bet, she will either win $5 with probability $\displaystyle \frac {18}{38}$ or lose $5 with probability $\displaystyle \frac {20}{38}$.

If W is her final winnings and X is the number of bets she makes, then, since she would have won 4 bets and lost (X-4) bets, it follows that

$\displaystyle W=20-5(X-4)=40-5X$

Hence $\displaystyle E[W]=40-5E[X]=40-5[4/(\frac {9}{19})]=-\frac{20}{9}$

but why is $\displaystyle E[X]=4/(\frac{9}{19})$?