Looked at the past 10 drafts. The team with the worst record only drafted first 1.5 times. The average pick of the worst team was 2.8 respectively.
Below is the list of where the worst team in the league was slotted after the lottery, and who they took with their picks:
Asuming: draft order is wiggins,randle,parker,exum
2013 Draft: #2 - Oladipo Randle
2012 Draft: #2 - Kidd-Gilchrist Randle
2011 Draft: #2 - Derrick Williams Randle
2010 Draft: #3 - Favors Parker
2009 Draft: #4 - Tyreke Evans Exum
2008 Draft: #2 - Michael Beasley Randle
2007 Draft: #4 - Conley Jr. Exum
2006 Draft: #4 - Tyrus Thomas (POR eventually ended up with Brandon Roy) Exum
2005 Draft: #2 - Marvin Williams Randle
2004 Draft: #1 - Dwight Wiggins
2003 Draft: #1 & #3 - Lebron & Melo (Cleveland and Denver tied for the worst record that year) Wiggins
If this was the draft order for the last ten years then the worst team would have the opportunity to pick
Wiggins 1.5x
Randle 6.5x
Parker 7x
Exum 10x