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Message: <blockquote data-quote="Mike Sullivan" data-source="post: 1005051" data-attributes="member: 9824">Okay, here's how it works:What happens when you get a critical hit?  You roll two d20's -- a specific one of them has to roll a hit, and the other one of them has to roll within the crit range.As everyone knows, the odds of two independent events both coming out in a particular way are equal to the odds of one of those events times the odds of another of those events.So, let's take a specific example:  A longsword, say, without the keen trait or the improved critical feat -- in other words, a fully modified crit of 19-20/x2.Now, let's say that the odds of hitting with his longsword is a variable we'll call m.  What are the odds of critical-hitting with that longsword?Well, it's the odds of rolling 19-20 on a d20, times the odds of hittimg (m).  The odds of rolling 19-20 are 10%.  So, the odds of critical hitting are .1m.Now, we know that the odds of hitting at all are = m.  We also know that a critical hit is, naturally, a hit as well.  Thus, we can calculate the odds of rolling a normal hit as m - .1m, or .9m.Thus, we can see that 10% of all hits are critical hits, and 90% are normal hits -- regardless of what AC we're attacking!  (But see below).  Naturally, both hits and critical hits are rarer with higher AC's, but the proportion between them is static.Then, we note that a critical hit for this longsword does x2 damage -- so if the damage we do (before crits) is the variable d, then 90% of the time, we'll do d damage, and 10% of the time we'll do 2d damage.  Thus, our expected damage for each hit is .9d + .1*2*d.  That gives us an expected damage of 1.1d.If you perform a similar calculation for every crit range, you'll find that this is true:If your weapon has a 20/x2 crit, then to get your expected damage including crits, multiply your expected damage before crits by 1.05If your weapon has a 19-20/x2 or a 20/x3 crit, then multiply by 1.1If your weapon has an 18-20/x2 or a 20/x4 crit, multiply by 1.15If your weapon has a 17-20/x2 or a 19-20/x3 crit, then multiply by 1.2If your weapon has a 15-20/x2 or an 18-20/x3, or a 19-20/x4, then multiply by 1.3If your weapon has a 12-20/x2 or an 18-20/x4, then multiply by 1.45One problem with the abstraction above: it doesn't take into account the situation where your roll to hit is higher than your crit thresh -- so, if you have a 12-20/x2 critical range, but you require a 15+ to hit, you can't just naively multiply your damage by 1.45 like you ordinarily would.  In that case, your crit threshold has been effectively reduced to 15-20, so you'd only multiply by 1.3.No, they don't.  They assume that the odds of hitting are the same between all the different weapons we're comparing, and they don't at all assume that you always confirm or threaten.If it helps any, what I'm doing here is not controversial.  This is about the 5th time I've posted these formulae on this board, and I've done similar stuff on RPG.net.  Nobody has ever found any errors in the methodology, and a variety of other people have come to the same numbers from first principles -- and I'm hardly the first person to come up with these numbers.  They are, to be totally blunt, correct.They don't necessarily tell the whole story -- no figure that's based on averages ever will -- but they're correct as far as they go.Yes, and the odds of hitting for that same scenario are 95%.  So, as you can see, 10% of all of the hits will be critical hits (10% of 95% being 9.5%).EDIT:  On a second reading of your post, I think I see the point of confusion.  What I am saying is that (in the scenario above), 10% of all HITS are critical hits.  That is not the same as saying that 10% of all ATTACKS are critical hits -- since no attack has a 100% chance to hit, as you correctly note, that means that there is always less than a 10% chance that a given attack is a critical hit.  However, when we're talking about differences between weapons, we don't want to talk about ATTACKS, we want to talk hits -- because adding AC's into the picture complicates the scenario, to not particular advantage.On the other hand, if we were discussing power attack, or two-weapon-fighting, or whatever, where the chance to hit is different in two seperate scenarios, then we do have to discuss attacks -- and when that's appropriate, I take it into account.  For an example of the rather more complicated analyses that this engenders, take a look at some of the several 3.5 Power Attack threads that have gone on recently -- in at least two of them, I worked out the math extensively.</blockquote>

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