How many battles could a Napoleonic war or ACW infantryman expect to survive?

Depends My math says 7 battles is a greater than 50% chance of being killed/wounded, and by 13 battles you are "statistically certain" to have been injured on a ACW battlefield.

I'll be using American Civil War stats, because I'm far more well-versed in that then Napoleonic Warfare. The major difference survival-wise is that ACW warfare almost never resulted in the actual destruction of the defeated army. In Napoleonic warfare cavalry riding down routed infantry was a fairly common occurrence, and may swing the casualty numbers. Conversely European advisors were near-universally aghast at the fact that ACW units would volley back and forth for hours rather than one side attempting to close with the bayonet. Since bayonet charges usually end in one side breaking and running before actual stabbing begins, it was widely believed at the time that prolonged firefights meant higher casualties. Ok that's the caveats out of the way. So how long did Johnny Reb/Billy Yank last on average?

The overall fatality rate for Civil War soldiers was 1 in 5. However 2/3 of these are deaths from disease, not strictly battlefield casualties. This site list all Killed in action (KIA) and Wounded in Action (WIA) for the war on both sides at 1.1 million. approximately 3.2 million men served on both sides. But this also isn't incredibly useful, as a man could be wounded multiple times, or wounded and later killed. Plus not all soldiers ended up in a battle.

The average casualties for an army in a battle also varies, from an average of about 6-8% for the winning side and 12-14% for the losing side. Of course this takes into account Prisoners as well, some of whom were presumably captured un-injured. Naturally some battles were also more bloody than others, with the top 10 battles of the war accounting for 17% of the war's TOTAL number of casualties Source. Even worse, some of these battles were horrifically one-sided! A union soldier at Gettysburg had a far higher chance of being killed or wounded than his confederate opposite number. Then there's the fact that some units in the hottest fighting would get massacred (100% dead/wounded in at least one CSA regiment in picket's charge, whereas other CSA regiments at Gettysburg had 1-2% casualties for the whole 3 days of battle!).

So what does all this mean? The best way to calculate the "war-wide average" seems to be taking the average win/loss casualty ratio and averaging them out. Which means 7% and 13% for winners/losers, averaged to 10% of all soldiers become casualties. So 10 guys in a notionally full-strength 100 man company, easy math. Per (Source) 400,000 soldiers became POWs. With 1.1 million casualties total and 400k POWs, we can asy that ROUGHLY 26% of battlefield "casualties" were actually POWs. (please check my math, I'm a history major not a mathematician!) So let's say 25% for sake of maths. So of our 10 battlefield casualties, 25% of them will be prisoners. Some of whom are wounded, but not all. Frankly I could find zero info on "wounded vs unwounded" POWs so I'm going to treat them as mostly unwounded for now. But realize that may make this number low.

So of our 10 guys 2.5 were captured. let's say 2 and call the half guy representative of "wounded POWS." so 8 guys out of 100 in any given battle are killed/wounded. So 8% casualties in every battle means that by battle 9 there is a 53% chance of you being killed/wounded, using "conditional probability. By battle 16 that number is about 74%. Of course this is VERY rough numbers. The Stonewall brigade and other units suffered higher losses than other units that were not continually relied upon to hold in tight places/attack against long odds. The odds also vary greatly between infantry and cavalry/artillery, your rank (any given general being more likely to be killed than any given private), your campaign region (west and east had different intensity of warfare) and the battles themselves. (A large battle has a higher chance your unit gets decimated, but also a higher chance you're only lightly engaged and lose very few men.)

Honestly I think doing the math for this on any group larger than a particular unit gives results too fuzzy for real use, as so much depends on the specific battles, and even parts of battles, a person was in. But this is better than nothing and I hope it helps!