Expected number of red balls drawn before drawing a white ball

[txy]

Homework Statement

There are 50 balls in a bag, all are the same except for the color. 4 of them are white while the rest (46) of them are red. Now randomly draw balls from the bag, one at a time, without replacement. What is the expected number of red balls that you will draw before you first draw a white ball?

Homework Equations

If a discrete random variable X takes only positive values, then its expectation
[tex]E(X) = \sum_{k=1}^{+\infty}P(X \geq k)[/tex]

The Attempt at a Solution

Let X be the number of red balls drawn before first drawing a white ball. I can't seem to find [tex]P(X \geq k)[/tex]. Can somebody give me a hint?

 

[jeppetrost]

Say it out loud :)
What is the probability, that you draw 1 or more red balls? It is of course the probability of the first ball being red.
What is the probability, that you draw 2 or more red balls?
Etc.
You should get a nice sum, which you will be able to write in a closed form.

 

[txy]

The probability of the first k balls is red is [tex]\frac{C_{46}^{k}}{C_{50}^{k}}[/tex]

This is so difficult to sum!

 

[tmccullough]

Is it OK for the answer to just be a sum? It might be the case that this doesn't have a better looking form than a sum.

 

[txy]

But the answer is 46/5 ...