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 or lose $5 with probability .

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

Hence

but why is ?