You nailed the reason why the average was wrong by speaking of the bell-shaped graphs, its more of a weighted average. The chance of not getting the ball you want after 1 pull is 1/2. After 2 pulls it is (50/100)*(50/99), the chance of you missing it the first time times the chance of you missing it the second time (99 because one ball is now out of the running). The chance of you pulling 49 balls out and still not getting the one you want is (50/100)*(50/99)*(50/98)*...*(50/51) which is about 2.9*10^(-9). It *could* happen, it just likely will not.

On a side note, this is my first node that's not a question! I'm so happy I could cry! Would that people would ask more math based questions so I could actually answer them ^_^. ~Adam