I played with the stats one time assuming applying for all permits available and tossing in a few tickets for random raffles. Even if you assume really low chance of drawing 0.5%+/- your odds of drawing a in 30-40 years is pretty good. Thats also about a $2500+ investment every year, which seems to increase about 5% a year as well. Doesn't take long to pay for a guided hunt up north with 100% chance of getting a tag.
I used to apply for a few states, MT, CO and ID, but I just can't justify the costs vs reward/odds to apply, personally. If you look at the total number of NR sheep tags given in all the L48, its 100+/- NR tags, (probably less, and some states are looking to limit NR even further). More than half of all NR tags are given out in WY and NV, with CO third. The rest of the states give a max of about 10 tags each, if that. I lost my desire to hunt bighorn sheep after I killed a few sheep in AK and spent a lot of time hunting up here. The hunts in the L48 just don't seem as wild as they used to be, but everyone has their own experience/expectations.
I hunted the UL in MT a couple times in the late 90's early 2000s, and at the time it was a really cool experience and would still be today. I would recommend that everyone have a chance to do that, as long as they go into it knowing its mostly likely going to be a camping trip. Your odds to kill one are about as good as your odds of drawing a tag, maybe slightly more.
I'm lucky enough to get all the sheep hunting I can handle each year in AK. My wife and I have both drawn 2 tags each in the last 12 years up here. "Odds" for all tags we drew ranged from 10% (one archery tag) and 2-3% on other draw tags. Someone has to draw, and I don't mind tossing in a few bucks a chance at a tag. Its funny that most people don't realize that OTC areas have better success rates than most draw areas.
Honestly, if you really want to hunt sheep. Make contacts/connections with sheep hunters, to the WSF and make friends, but most importantly get a good paying job (or side work) and start saving. The best thing I ever did was move to Alaska, and I wish I would have done it sooner.
I get there are lots of people that want to collect all the species, and that love hunting BH sheep. Everyone has their own personal expectations/experiences. You can also do both, apply, save and hunt sheep in AK/Canada. Its an expensive game anyway you look at it.
Also note that a lot of guys that have drawn a couple tags, were also in on the ground flow of point systems. Coming in 20 years late is a huge headwind to buck.