Your best chance to draw would probably be AZ or possibly NV. These states seem to have the most sheep. But remember, you asked for best chance and not least expensive. A NR hunt license will cost you $160 per year. You will need that just to apply. If you are not drawn, you are only charged the $7.50 application fee. If you are lucky enough to draw, the tag will run you $1,815.
California's system sounds like it heavily favors the max point holders. In AZ there are typically some 70 permits for desert bighorn. Of those, only 20% or 14 tags are guaranteed to go to the max point holders. The rest are open for anyone to draw. Although a lot of the remaining tags go hunters with higher points, there are always some sheep tags that find their way to hunters with only 5 points - or even first time applicants. It is truly like winning the lottery.
This year there are approx. 160 hunters with max points (25). With an attrition rate of about 25 per year due to drawn tags, plus another 2-3 hunters that either dir or give up, we will burn thru our current max hunters in about 5-6 years. Then those with 24 points (currently) will be the next group. There are over 200 of them, but about 10 draw or give up/die each year.
The AZ system seems to be a little more fair than CA, but it is still long odds if you didn't get into the game early. Statistically, I will begin to have greater than a 1% chance of getting drawn by the age of 65.