Illustrator :: Should Use RGB Or CMYK

I am currently designing a billboard on Illustrator. The background is black on the screen. I used CMYK  0 0 0 100.  But when I save it as pdf or export is as JPG, it becomes greyish in pdf/JPG. Why is that? On the other hand, if I use RGB in Illustrator (and not CMYK) and use RBG 0 0 0, what I get in the pdf or JPG is what I see on the screen, that is black. The colour does not change in thsi case from illustrator (black) to pdf/JPG (grey).
 
Should I use RGB or CMYK? Is there a way in which I can use CMYK and secure the same colours for my illustrator file and pdf file?  Using RGB solved the problem but do printers accept RGB? RGB seems to secure consistency between pdf, JPG and illustrator but not CMYK.