On average, how many times must I roll a dice until I get a $6$? On average, how many times must I roll a dice until I get a $6$?
I got this question from a book called Fifty Challenging Problems in Probability. 
The answer is $6$, and I understand the solution the book has given me. However, I want to know why the following logic does not work: The chance that we do not get a $6$ is $5/6$. In order to find the number of dice rolls needed, I want the probability of there being a $6$ in $n$ rolls being $1/2$ in order to find the average. So I solve the equation $(5/6)^n=1/2$, which gives me $n=3.8$-ish. That number makes sense to me intuitively, where the number $6$ does not make sense intuitively. I feel like on average, I would need to roll about $3$-$4$ times to get a $6$. Sometimes, I will have to roll less than $3$-$4$ times, and sometimes I will have to roll more than $3$-$4$ times.
Please note that I am not asking how to solve this question, but what is wrong with my logic above.
Thank you!

The experiment is: roll a die until you get a six.
The median is $3.8:$ That means that half the time when you perform this experiment you will get your six in under $3.8$ rolls and half the time you won't.
The expected value is $6.$ This means that if you performed the experiment a hundred times and added all the rolls from each experiment together you should get around $600$ total rolls. So one could get the same total by assuming we had $6$ rolls in each experiment. 
Think of it like this: although you have a $50\%$ chance of it taking less than $3.8$ rolls there are still gonna be a lot of times where it takes $8, 9, 10$ or more. Those high numbers are going to skew your expected values and leave you with an average of $6.$
The question you asked is a good one and goes right to the heart of what we mean by expected value.
