Expected number of cards drawn before drawing a $4$ or $5$ I'm working on the following problem:

Compute the number of expected cards drawn from a standard 52 card
  deck (without replacement) until a $4$ or $5$ is drawn.

I tried to model it using a geometric distribution, but am running into problems since the probability of drawing a $4$ or $5$ increases with each successive card drawn. Could this problem be approached using Markov Chains?

For this problem, we can develop a formula for $P(x=k)$.  Where the variable x represents the draw on which we obtain a success.  $$P(x=1)=2/13$$ $$P(x=2) = 44/52*8/51$$ $$P(x=3)=(44/52)*(43/51)*(8/50)$$continuing in this fashion, we obtain the formula: $$P(x=k)=8*(44!/52!) \frac{(52-k)!}{(45-k)!}$$  Thus, we can use the definition of expected value: $E(x)=\sum_{k=1}^{45}k\cdot P(x=k)$  Unfortunately, I know no method to solve this sum. So of course, I resorted to wolfram which spit out $53/9$.  Thus, $$E(x) = 53/9 = 5.888...$$
