I did some math on how likely you are to get your desired role. I read an analysis a while back that broke down which roles were most desired and the numbers for each role were close enough that I used an even distribution.
So...
If you are first pick you have 100% of picking your role
if you are second pick you have 80% chance
If you are third pick you have 64% chance
If you are fourth pick you have 48% chance
If you are last pick you still have 32% chance
Averaging all of these and weighting appropriately gives you overall a 64.8% chance of getting your desired role in any game.
Additionally, I have tracked my actual role over the last 94 ranked games. I have gotten my desired role 70.2% of these games. My weakest role is jungle, which I have played only once this ranked season.