First off, note that I will assume the system completely RAW and core rules only, since there are a thousand possible houserules and we can't consider them all.
Well, the Barbarian would be stupid if he didn't have Improved Ini either, and otoh there's a lot to take for the Nomad before Lightning Reflexes really make sense. So assuming sensible builds for both, they are going to have pretty much the same Ini mod. So both have a 50-50 chance of winning Ini.
So what could the builds credibly look like? I'm not trying to tailor the characters for this encounter, but to figure out sensible and survivable builds that you might actually encounter in play.
Raw stats for both: 18, 16, 16, 14, 14, 12 (incidentally, in this order. This includes stat upgrades until level 10). A Cimmerian Barb will have Str 20 and Int 12.
A Nomad worth his salt will certainly have Mounted Combat, Ride By Attack, Spirited Charge. A Hyrkanian is also likely to have Mounted Archery and Far Shot. That's 3-5 feats that will easily decide the combat in a scenario where he can make a mounted charge, but won't help him at all in a melee on foot.
If he's clever he'll have Improved Initiative. Let's also give him Weapon Focus: Scimitar. And Power Attack.
The Barb is just as likely to have Improved Ini and PA. It's difficult to say if he's going to have other relevant feats; for instance he might have Improved Grapple and even Crushing Grip. Fighting Madness is also likely, but we'll leave it out for now.
Let's assume that both have the same armour - but the Barb's DR is going to be 1 better because of his class feature.
Combat stats:
Nomad: Attack +15/+10 (Scim), DD 18, PD 19, DR6, HP84, AP6*, Ini +14
Barb: Attack +15/+10, DD 20, PD 18, DR7, AP8, Ini +14
*) This is the first big If. If one fighter can pierce the other's armour, but not vice versa, that's going to be a HUGE advantage for one of them.
Let's give the Nomad a Scimitar and the Barb gets a Broadsword. No 2-handed weapons for either. Both can draw a Light offhand weapon if they want to. Maybe both have ITWF even.
Maybe there are shields available, that would be a certain advantage for the Nomad.
Now they roll for initiative, it's a 50-50 chance for both.
If the Barb wins Initiative: Nomad is flat-footed, cannot Dodge or Parry, Barb charges (+2) with 2H-PA -9, can only miss on a natural 1, pierces armour, inflicts ~d10 + 7 + 18 - 3 on a hit = 95% to kill on first attack. Hell, even if the Nomad for some reason has DR8 and the Barb doesn't pierce it, there's still a 70% chance he'll roll at least 20 damage!
If Nomad wins Ini: Barb is flat-footed but can use Uncanny Dodge. Nomad charges in, gets Nomad Charge bonus but may not be able to pierce armour. So if he pumps up his Power Attack to make Massive Damage possible, he's unlikely to overcome the Barb's unimpeded Defense. If he plays it safe to ensure a hit, the Barb will shrug off a good part of the damage and just soak the rest.
Then unless one of them has another ace up the sleeve (i.e. special maneuver), it's going to be just a war of attrition and the side with better average damage or simply more luck will win.
Of course there's still the chance for a Freak Roll, like a Critical Hit or a lucky Power Attack. Either way, in this sub-scenario the forces are pretty evenly matched, since both have near-identical attack, defense and hit points. For simplicity's sake let's just say it's 50-50 here.
Special Attack: if one combatant has a suitable Special Attack, that can also change the duel drastically. Maybe the Nomad has Improved Sunder and manages to chop up the Barb's sword. But maybe the Barb has Crushing Grip and simply grapples and squeezes the Nomad like Elmira smothers a Tiny Toon. This is really difficult to say.
The Nomad might have the best chances if he also had Improved Critical:Scimitar, that would give him a 30% chance of rolling a threat, but alas, even a confirmed crit with a very good damage roll (2*(d8+4+PA)-DR) may not be enough to cut the battle short.
So to sum up:
If the Barb wins Initiative, which will happen 50% of the time, the fight will be over as soon as it has begun and the Nomad will be a red patch on the ground.
If the Nomad wins Initiative, also at 50%, there's again a rough 50% for each fighter to ultimately carry the victory. 50% times 50% is 25%.
So the final odds are 75-25 in favour of the Barbarian.
Of course this scenario vastly changes if you apply house rules and nerf 2HPA. Then the odds would probably be a LOT closer. But as I said, in this kind of simulation we have to stick to the RAW to keep it comparable.
