"Ouch?!" - improved crit or +2 damage?

[JamesDJarvis]

if using a longsword a +2 to damage is better then improved ciritcal for average damage inflicted. . You are likely to inflict 131% of the damage with a +2 to damage compared to an improved critical.

50% chance to hit with improved critcal means 50% miss,40% of blows will be normal (average 4.5) and 10% will be critical (since only half of the 20% threat will be confirmed) for damage figures of (40*4.5 + 10 * 9)/2 = 135 pts

50% chance to hit with +2 damage means 50% miss, 45% of blows normal (for 6.5 average) and 5% are likelt to be 13 pts of damage.
damage is (45*6.5+ 13*5)/2 = 178 pts

 

[SBMC]

Keep it simple

Nice number crunching everyone! Quite impressive - though I go for a simpler approach with the question at hand:

IMO with the plethora of fighters I have played - if you can use it every single attack roll take it. Both of these are "on" every attack role but only one is guaranteed to pay off 100% of the time - provided ya hit at all - but if you critical you hit anyway.

Weapon Specialization like Weapon focus is used every single attack roll without exception. As such I would say get the +2 first; then get Improved Critical later. Especially at low levels where 2 points of damage can make a bigger difference.

I have made tons of criticals with the Improved Critical feat in my portfolio - but again - the +2 every single hit...

Simple math without real math:
Think about it - if you have a long sword with Improved Critical you have a 4 in 20 chance of making a critical hit - 20% chance per dice roll, without accounting for the need to confirm (which actually drives that percenage down dependant upon you total BAB and modifiers), versus 100% chance of inflciting the 2 points of damage. Plus it gives ya access to inflict 4 points at 12th level (provided your a fighter)

Logic, perhaps supported or not by numbers (which goes all over the place depending on your logic), IMO points towards the guarantee not the gamble.

SBMC

 

[orsal]

Nice number crunching everyone! Quite impressive - though I go for a simpler approach with the question at hand:

IMO with the plethora of fighters I have played - if you can use it every single attack roll take it.

Not a good rule of thumb. Improved Critical, when it works, gives you much more than the +2. Any time there's a tradeoff between a small "sure" bonus and a large possible bonus, you need to pay attention to the probabilities. In this case, you make the right decision by your rule, but what if you were offered the choice between only +1 damage or doubling your chance at a critical? In that case, it would depend on weapon type and damage bonuses -- with a good enough weapon the more frequent criticals would be worth it. That extra 1 HP damage might be a sure thing with every hit, but it's rarely going to be the difference between life and death for your adversary. The critical, when it does happen, is more likely really to matter.

 

[SBMC]

Not a good rule of thumb. Improved Critical, when it works, gives you much more than the +2.

Key words - "when it works". You see I am not talking about one hit as you are when referring to the critical - with only a 20% chance (referring to my example above in my previous post) to hit; that means, effectively, that I will roll 16 times before I make a critical hit - and taht is if everythign is noce an evenly distributed over a nice neat bell curve within the constraints of 20 rolls. 16X2 is 32 points of damage.

When having improved crit with a particualr weapon sure the numbers may be against me regarding probabilities; but it is more probable than not per individual roll that it will not be a crit.

I am a student of game theory and the way I look at it is from a micro not a macro standpoint: When you play craps in Las Vegas; it is that roll that matters not the one before it or after it. Each roll is a stand alone event with it's own probabilities. If you played craps and got an automatic +2 (towards whatever would be the better number) to every roll you made would you take that or take a +4 every 16th roll?

I roll a million times the averages work out - I know that - and that is what you all are talking about - but you hafve to roll many times for those probabilities to come about be them in your favor or not (as in, for an extreme example, the first 30,00 rolls are crits, the next are not, or vice versa). I'll still take the guarantee. It's like buying insurance; you give something up (money) but whenever you need it, no matter what, you know it will be there. Sure "maybe" I could hit it big in the stock market with that money and end up ahead in the long run...would you take that chance? Depends on your own desires.

Now the extra damage is especially important at 4th level when HP's are low yet a long sword still does 1d8; 2 HP makes a difference. At 8th level; less of course...but it is still 2 HP

Besides this has worked for me; and I always take both Weapon Spec and Improved Crit over the levels anyways when playing a fighter.

Any time there's a tradeoff between a small "sure" bonus and a large possible bonus, you need to pay attention to the probabilities. In this case, you make the right decision by your rule,...

And IMO (noting it is an opinion) it always is; I am not a risk taker (as a person; I play the opposite in the game though!)

...but what if you were offered the choice between only +1 damage or doubling your chance at a critical? In that case, it would depend on weapon type and damage bonuses -- with a good enough weapon the more frequent criticals would be worth it. That extra 1 HP damage might be a sure thing with every hit, but it's rarely going to be the difference between life and death for your adversary. The critical, when it does happen, is more likely really to matter.

If I am faced with that I'll deal with it. At lower levels I would still take the 1 HP of damage.

The big hit is always great; buty again; the +2, hit after hit after hit is there regardless of the averages with rolls and crits. I could roll 23,000 times and not get a crit; but still get the +2 every time I hit. Of course later down the road the averages work in there as mentioned above; but that is my point - later the crit happens, now the +2 happens.

 

[Scion]

given the nature of craps vs the nature of d&d I'd have to say that the +2/+4 comparison is completely wrong

+2 change in craps is 'huge'. You'll never roll a bad number and the vast majority of the time you will win on the first roll. A +4 at that point will almost never matter.

However, in d&d there are a lot of factors that add onto damage. Weapon damage + str + spell bonuses + feat bonuses + item bonuses + who knows what else.

Hence looking at the overall average damage. Given the sheer number of rolls made in d&d over the course of a level things will be very close to the average. If you are worrying about a choice between two combat feats for the damage that they cause even after a mere 40 rolls (a couple of days of combat at the very most really) you are incredibly likely to get the most common value.

Extending the crit range does actually change your average damage.


But even with all of that said there is nothing wrong with going for the sure thing. it is completely understandable. Even if it could be shown that improved crit was better 100% of the time on average there would still be nothing wrong with picking the sure thing. In fact, depending on the weapon I was useing and the character concept that would choose the feat more than the numbers would.

 

[orsal]

Key words - "when it works". You see I am not talking about one hit as you are when referring to the critical - with only a 20% chance (referring to my example above in my previous post) to hit; that means, effectively, that I will roll 16 times before I make a critical hit - and taht is if everythign is noce an evenly distributed over a nice neat bell curve within the constraints of 20 rolls. 16X2 is 32 points of damage.

With only a 20% chance to hit, you will only get that +2 damage about 20% of the time. That is why I put your word "sure" in quotes.

Anyway, you're implicitly conceding my point -- you've retreated to the same kind of calculation that I (and others) did earlier, only our probabilities were correct.

Oh, and "bell curve"? A single die has uniform distribution... nowhere near a bell curve.

I am a student of game theory and the way I look at it is from a micro not a macro standpoint: When you play craps in Las Vegas; it is that roll that matters not the one before it or after it. Each roll is a stand alone event with it's own probabilities.

Its own probabilities... and its own payoffs. If I offer you a bet that you will win 2 times out of 3, but that will pay you $1 if you win while costing you $3 if you lose, you'd be a fool to take it. The disparity in the payoff doesn't make up for the disparity in the probability of winning.

If you played craps and got an automatic +2 (towards whatever would be the better number) to every roll you made would you take that or take a +4 every 16th roll?

The automatic +2, of course... but if instead of +4 one time in 16, it was +10 one time in 16, while the +2 was only +2 one time in 4, that changes everything. I reiterate my previous point: you need to consider both the probability and the payoff to make a smart choice.

I roll a million times the averages work out - I know that - and that is what you all are talking about - but you hafve to roll many times for those probabilities to come about be them in your favor or not

And I expect my fighter to be making scores of attack rolls over the course of the campaign, so I'm going to take the long-term view.

It's like buying insurance; you give something up (money) but whenever you need it, no matter what, you know it will be there. Sure "maybe" I could hit it big in the stock market with that money and end up ahead in the long run...would you take that chance? Depends on your own desires.

Depends on my situation. The idea behind insurance is that the customer, while taking a policy with a negative expected value, will get the money in the contingency that he'll need it the most. A concave down utility function -- the same money is more important to him in adversity than in prosperity. I don't see that applying here.

I could roll 23,000 times and not get a crit;

...that's so unlikely that you can reasonably ignore the possibility. If you want to say "but there is that nonzero probability, however small".. I'll point out that there is also a nonzero probability that in your 23,000 rolls you won't even get a *hit*, so the +2 damage would then come to naught.

 

[SBMC]

With only a 20% chance to hit, you will only get that +2 damage about 20% of the time. That is why I put your word "sure" in quotes.

Ho, hum...20% chance to make a critical; not an actual hit.

Anyway, you're implicitly conceding my point -- you've retreated to the same kind of calculation that I (and others) did earlier, only our probabilities were correct.

I had already conceded in earlier posts that my choices could very well, over time or not, be less advantageous than others - and that they are indeed not based on numbers but by preference - in response to the original post that started the thread.



"Retreat": Interesting choice of words; not intended to insult but intended to take the other readers to a particular frame of thought before you continue. A move that is really unnecessary. And when you toss numbers at me exactly how would you like me to respond? And what exactly is wrong with saying 4 out of 20 equals 20%?


And you numbers are not correct; you forgot important variables in your formulas. You could knowingly ignore of course - you'll see them below.

Oh, and "bell curve"? A single die has uniform distribution... nowhere near a bell curve.

Read the post again - I said 16 rolls directly and implied 20 rolls. Looks like you were trying to pull the words apart and not listen to the message - a message that does not even argue to your points directly.

Its own probabilities... and its own payoffs. If I offer you a bet that you will win 2 times out of 3, but that will pay you $1 if you win while costing you $3 if you lose, you'd be a fool to take it. The disparity in the payoff doesn't make up for the disparity in the probability of winning.

And that points to the fact you did not read my post; simply attempted to make me a victim of semantics and that you really don't know what I am talking about - I stated CLEARLY - MICRO. Meaning each individual iteration NOT the combination thereof. You intentionally ignored that point in order to make your own; why not make your own not at my expense?

The automatic +2, of course... but if instead of +4 one time in 16, it was +10 one time in 16, while the +2 was only +2 one time in 4, that changes everything. I reiterate my previous point: you need to consider both the probability and the payoff to make a smart choice.

That +2 would actually be 80% of the time. The +4 would be 20% of the time. Fine; change the example; +10 20% of the time - my point stays the same and it is a point of rational choice in addition to being backed in micro terms - not in all cases but this is my personal opinion you are touting as if I had stated it is the way everyone should go and it was backed with hard evidence - recall the multiple insertions I made of "IMO" or "opinion" in my posts? Most likely not since it is clear you did not read it; yet chose to reply to it; I'll leave it at that; anyone here with an maturity sees where I am going.

And I expect my fighter to be making scores of attack rolls over the course of the campaign, so I'm going to take the long-term view.

And over the course of an entire campaign you could easily get BOTH feats as a fighter. Not to mention you cannot get Improved Critical until 8th level - THAT IS ONE OF THE MISSING FACTORS.

In reality all of your calculations are irrelevant from level 4 (where you can get Weapon Specialization) to level 7. Then at level 8 you can gain Improved Critical as well. If you took both y level 8 the point of this is all moot.

Thus; to be realistic; your calculations should not even start until level 8 is reached.

But then again why is this even important or relevant? Well you asked for it:

I note you ignored my point regarding 2 HP of damage at lower levels versus higher levels

You are all calculating the probability of a particular roll(s) over time and iterations. In mathematical terms the roll, in reality, is not relevant as it is in gambling - the benefit of the roll is. Why? Because it is not, as all of your calculations state, a 1 or 0 ending result.

I am not going to spend my time doing this but feel free if you want to:
You need to build a scale - a scale that determines the value of 2 HP of damage versus the critical damage taken and the probabilities.


Ya see at 8th level, for example, the average fighter has 45 HP (no CON modifiers).
2/45 = 4.444444%


At 9th level: 50 HP
50/2: 4.00%


At 10th level: 55 HP
55/2: 3.636363%


Ya'all get my point here? I hope so after seeing the calculations you have all done. The lower your level the more valuable that +2 is given the HP's of the creatures you are facing. Versus Improved Crit; who knows; you do the math.

Also - You all currently have an infinite number of iterations possible yet wish to communicate the result within the confines of averages within the game.

Thus you should be utilizing the average attack rolls per encounter times the average encounters to determine this - not just open ended probability. Thus your parameters are fixed within averages as are the probabilities per roll: Within the confines of X rolls I have he probability of rolling a critical…

You will see how that changes things drastically. Like I said I am not doing this out (and to note for Mr. Orsal; Yes I can; quite well) as I have no interest in doing so - especially since I know all of you who love mathematics will start to work on that ASAP and do I for me! :--)

Depends on my situation. The idea behind insurance is that the customer, while taking a policy with a negative expected value, will get the money in the contingency that he'll need it the most. A concave down utility function -- the same money is more important to him in adversity than in prosperity. I don't see that applying here.

Really - explain further without regurgitating a textbook word for word for me would you please; in your own words. And please explain this to me from the Insurance companies point of view? Oh…you would no want to do that since it would support the point I was making which again, like many of my earlier statements you commented on in a similar fashion, was just an example to foster a comparison of perspective not mathematics - and unless you are completely daft you knew that.

...that's so unlikely that you can reasonably ignore the possibility. If you want to say "but there is that nonzero probability, however small".. I'll point out that there is also a nonzero probability that in your 23,000 rolls you won't even get a *hit*, so the +2 damage would then come to naught.

"Come to naught" - then so would the Critical; a wash numericly.

Unlikely? Your, and everyone elses, basis is on probabilities. This is no more probable than any other set combination of results that is similar. No matter how you do it the numbers will, in the long run, represent a bell curve and everyone here knows it - and we are all talking about actual hits not necessarily rolls of the die - again; attacking the words; for what purpose? How all that comes to be over time and iterations is pure chance - read game theory.

 

[orsal]

Read the post again - I said 16 rolls directly and implied 20 rolls. Looks like you were trying to pull the words apart and not listen to the message - a message that does not even argue to your points directly.

What does that have to with the difference between a uniform distribution and a bell curve?

And that points to the fact you did not read my post; simply attempted to make me a victim of semantics and that you really don't know what I am talking about - I stated CLEARLY - MICRO. Meaning each individual iteration NOT the combination thereof.

But the outcome of a particular attack is random. Where we disagree is how best to evaluate an option with uncertain outcomes. I go with expected value, which takes into account both the values of the various possible outcomes and their probabilities.

Oh, and to go back to the original question... if you're deciding what feat to take, the decision will affect many attack rolls over the course of your adventuring campaign. Not just one. Of course, by my standard (expected value of damage), it doesn't make a difference.

Most likely not since it is clear you did not read it; yet chose to reply to it;

I certainly did read it, but it is very difficult to argue with someone who repeats his own assumptions about what his interlocutor has or has not read. If you persist in making such allegations I shall not continue to argue, since I see no point in it; such aspersionsmake it very difficult to have a remotely respectful discussion.

And over the course of an entire campaign you could easily get BOTH feats as a fighter. Not to mention you cannot get Improved Critical until 8th level - THAT IS ONE OF THE MISSING FACTORS.

I answered the question asked by the OP, which was phrased as an either-or. There are plenty of reasons why you might be concerned to take one or the other -- you might be thinking medium-term (before your next feat-gaining level), or might have other feat choices selected and just trying to fill one more slot.

Ya see at 8th level, for example, the average fighter has 45 HP (no CON modifiers).
2/45 = 4.444444%

At 9th level: 50 HP
50/2: 4.00%

At 10th level: 55 HP
55/2: 3.636363%

All small percentages. In other words, that extra +2 won't fell them, unless there have been other hits. Therefore, it is the aggregate of many hits rather than a single roll that matters.

Oh, and the fighter's hit points aren't relevant; the monster's HP are. And that's more variable. Most adventuring parties will encounter both large groups of low-HP opponents and individual high-HP opponents.

Ya'all get my point here? I hope so after seeing the calculations you have all done. The lower your level the more valuable that +2 is given the HP's of the creatures you are facing. Versus Improved Crit; who knows; you do the math.

On that we agree.

Thus you should be utilizing the average attack rolls per encounter times the average encounters to determine this - not just open ended probability. Thus your parameters are fixed within averages as are the probabilities per roll: Within the confines of X rolls I have he probability of rolling a critical…

The advantage to probability theory is that it can identify some parameters that capture the key aspects of a distribution. I know enough probability theory to use them.

Really - explain further without regurgitating a textbook word for word for me would you please; in your own words. And please explain this to me from the Insurance companies point of view?

From the customer's perspective: "My friend the probability geek tells me that insuring my house against natural isn't worth it, because the probability that I'll collect is only 2%, but the policy costs 3% of what it will pay me. In his words, the expected payoff is only half the cost of the policy. Big deal. The cost of the policy doesn't mean a lot to me -- I make enough money that, barring unforeseen catastrophes, the cost of insurance won't substantially affect my standard of living. But that unforeseen catastrophe worries me. IThe affect on my standard of living of the cost of the policy is less than 1% the affect of losing my house to flooding. So if I think in terms of standard of living, rather than money, it is worth it."

From the company's perspective: "The expected value is positive for us. Since we will insure many policies with the same terms, our long-term average payoff per policy will be approximately the expected payoff. So we use the expected payoff to decide that this is a good policy to offer."

Oh…you would no want to do that

Guess again. Once again, please let your interlocutors decide for themselves what they do and do not want to do.

since it would support the point I was making

Unless you want to argue that the fighter's situation is analogous to the homeowner's, no, I don't see how it supports your point. I don't see making a single hit with neither a critical nor a damage bonus beyond your strength modifier is equivalent to losing your house to flooding.

"Come to naught" - then so would the Critical; a wash numericly.

Which is why I began *my* calculations with a hit -- since neither feat under consideration affects the probability of getting that hit, we can ignore it. I was merely pointing out that your description of a bonus that affects every attack was inaccurate incontext. Tangential to the point of discussion, so I see no need to repeat it too often, but worth pointing it out once.

No matter how you do it the numbers will, in the long run, represent a bell curve

Which numbers do you have in mind? When you first mentioned a bell curve, I understood you were talking about attack rolls, and as I noted above, they have a uniform, not bell-shaped, distribution. It is completely unclear from the context which random variable you are referring to here. Please clarify if you wish me to decide whether I agree that the distribution is bell-shaped.

 

[werk]

That's awesome how I can't read the yellow font at all...
(keep in mind, not everyone uses your color scheme.)

Ho, hum...20% chance to make a critical; not an actual hit.







I had already conceded in earlier posts that my choices could very well, over time or not, be less advantageous than others - and that they are indeed not based on numbers but by preference - in response to the original post that started the thread.

Edit: can actually see it in quote box!

 

[SBMC]

First - most people when they respond include most of the post so that readers can see the context

Quote:
Originally Posted by SBMC

Read the post again - I said 16 rolls directly and implied 20 rolls. Looks like you were trying to pull the words apart and not listen to the message - a message that does not even argue to your points directly.


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What does that have to with the difference between a uniform distribution and a bell curve ?

Lets go over this again:
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I posted, in response to one of our posts:

Key words - "when it works". You see I am not talking about one hit as you are when referring to the critical - with only a 20% chance (referring to my example above in my previous post) to hit; that means, effectively, that I will roll 16 times before I make a critical hit - and that is if everything is nice an evenly distributed over a nice neat bell curve within the constraints of 20 rolls. 16X2 is 32 points of damage.

Noting that I also stated in the same post:

When having improved crit with a particular weapon sure the numbers may be against me regarding probabilities; but it is more probable than not per individual roll that it will not be a crit.

I am a student of game theory and the way I look at it is from a micro not a macro standpoint: When you play craps in Las Vegas; it is that roll that matters not the one before it or after it. Each roll is a stand alone event with it's own probabilities. If you played craps and got an automatic +2 (towards whatever would be the better number) to every roll you made would you take that or take a +4 every 16th roll?

I roll a million times the averages work out - I know that - and that is what you all are talking about - but you hafve to roll many times for those probabilities to come about be them in your favor or not (as in, for an extreme example, the first 30,00 rolls are crits, the next are not, or vice versa). I'll still take the guarantee. It's like buying insurance; you give something up (money) but whenever you need it, no matter what, you know it will be there. Sure "maybe" I could hit it big in the stock market with that money and end up ahead in the long run...would you take that chance? Depends on your own desires.

Now the extra damage is especially important at 4th level when HP's are low yet a long sword still does 1d8; 2 HP makes a difference. At 8th level; less of course...but it is still 2 HP

Besides this has worked for me; and I always take both Weapon Spec and Improved Crit over the levels anyways when playing a fighter.

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You replied with:

Quote
Originally Posted by SBMC
Key words - "when it works". You see I am not talking about one hit as you are when referring to the critical - with only a 20% chance (referring to my example above in my previous post) to hit; that means, effectively, that I will roll 16 times before I make a critical hit - and that is if everything is nice an evenly distributed over a nice neat bell curve within the constraints of 20 rolls. 16X2 is 32 points of damage.


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With only a 20% chance to hit, you will only get that +2 damage about 20% of the time. That is why I put your word "sure" in quotes.

Anyway, you're implicitly conceding my point -- you've retreated to the same kind of calculation that I (and others) did earlier, only our probabilities were correct.

Oh, and "bell curve"? A single die has uniform distribution... nowhere near a bell curve.

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Then I replied with

Quote:
Originally Posted by orsal
Oh, and "bell curve"? A single die has uniform distribution... nowhere near a bell curve.
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Read the post again - I said 16 rolls directly and implied 20 rolls. Looks like you were trying to pull the words apart and not listen to the message - a message that does not even argue to your points directly.

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Now your last reply:

Originally Posted by SBMC
Read the post again - I said 16 rolls directly and implied 20 rolls. Looks like you were trying to pull the words apart and not listen to the message - a message that does not even argue to your points directly.
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What does that have to with the difference between a uniform distribution and a bell curve?

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So lets explore this one:

#1 I state that you would effectively roll the dice 16 times before making a critical (eluding to rolls 17-20 being crits)

#2 You reply that that is irrelevant - " A single die has uniform distribution"

#3 I state that I was talking about effectively rolling the dice 16 times before making a critical (eluding to rolls 17-20 being crits)

#4 You reply "What does that have to with the difference between a uniform distribution and a bell curve?"

Who is talking about a uniform distribution and/versus a bell curve? I did not.
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Quote:
Originally Posted by SBMC
And that points to the fact you did not read my post; simply attempted to make me a victim of semantics and that you really don't know what I am talking about - I stated CLEARLY - MICRO. Meaning each individual iteration NOT the combination thereof.


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But the outcome of a particular attack is random. Where we disagree is how best to evaluate an option with uncertain outcomes. I go with expected value, which takes into account both the values of the various possible outcomes and their probabilities.

Oh, and to go back to the original question... if you're deciding what feat to take, the decision will affect many attack rolls over the course of your adventuring campaign. Not just one. Of course, by my standard (expected value of damage), it doesn't make a difference.

Again - I say that you should read more on game theory. This is the exact same idea as gambling in part with the exception that there is not a finite capacity for the "player" or the "house" (which is one reason I suggested using average encounters and average attacks)

Let me rephrase:

Each individual roll is individual unto itself- put them together and you have a curve. Waiting for that curve to form is where I do not believe that Improved Critical outshines Weapon Specialization - an immeasurable, intrinsic point.
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Quote:
Originally Posted by SBMC
Most likely not since it is clear you did not read it; yet chose to reply to it;


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I certainly did read it, but it is very difficult to argue with someone who repeats his own assumptions about what his interlocutor has or has not read. If you persist in making such allegations I shall not continue to argue, since I see no point in it; such aspersionsmake it very difficult to have a remotely respectful discussion.

How can you say you read my post based upon your responses? Based upon the cherry picking of statements taken out of context? Your content is not flawed necessarily - it is the framwwork with which I take issue.
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Quote:
Originally Posted by SBMC
And over the course of an entire campaign you could easily get BOTH feats as a fighter. Not to mention you cannot get Improved Critical until 8th level - THAT IS ONE OF THE MISSING FACTORS.
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I answered the question asked by the OP, which was phrased as an either-or. There are plenty of reasons why you might be concerned to take one or the other -- you might be thinking medium-term (before your next feat-gaining level), or might have other feat choices selected and just trying to fill one more slot.

See below
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Quote:
Originally Posted by SBMC
Ya see at 8th level, for example, the average fighter has 45 HP (no CON modifiers).
2/45 = 4.444444%
At 9th level: 50 HP
50/2: 4.00%

At 10th level: 55 HP
55/2: 3.636363%


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All small percentages. In other words, that extra +2 won't fell them, unless there have been other hits. Therefore, it is the aggregate of many hits rather than a single roll that matters.

Oh, and the fighter's hit points aren't relevant; the monster's HP are. And that's more variable. Most adventuring parties will encounter both large groups of low-HP opponents and individual high-HP opponents.

What your telling me is that an additional, on top of normal damage 4% of total HP is irrelevant? Over rounds, perhaps multiple attacks? Will 2 points fell them - of course not. Will it be 2 points closer to felling them yes; how many non critical single attacks have you seen kill an opponent over 5th level?

I used the fighters hit points as an example only; replace it with whatever you choose and do the math. I think most rational folks get the point. This is another example of you taking things out of context and ignoring the obvious - targeting choice of words and not content.

As far as encounters go; that is obvious and known to all.
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Quote:
Originally Posted by SBMC
Ya'all get my point here? I hope so after seeing the calculations you have all done. The lower your level the more valuable that +2 is given the HP's of the creatures you are facing. Versus Improved Crit; who knows; you do the math.
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On that we agree.

And thus you made my point just above - that you stated as being " All small percentages. In other words, that extra +2 won't fell them, unless there have been other hits. Therefore, it is the aggregate of many hits rather than a single roll that matters."

My point was, and I quote myself:

Ya see at 8th level, for example, the average fighter has 45 HP (no CON modifiers).
2/45 = 4.444444%

At 9th level: 50 HP
50/2: 4.00%

At 10th level: 55 HP
55/2: 3.636363%

Ya'all get my point here? I hope so after seeing the calculations you have all done. The lower your level the more valuable that +2 is given the HP's of the creatures you are facing….

There was more to follow in that section I posted; but not relevant here at the moment.
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Quote:
Originally Posted by SBMC
Thus you should be utilizing the average attack rolls per encounter times the average encounters to determine this - not just open ended probability. Thus your parameters are fixed within averages as are the probabilities per roll: Within the confines of X rolls I have he probability of rolling a critical…

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The advantage to probability theory is that it can identify some parameters that capture the key aspects of a distribution. I know enough probability theory to use them.

And is this actually incorporated into your numbers or not? If it is my apologies; if not?
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Quote:
Originally Posted by SBMC
Really - explain further without regurgitating a textbook word for word for me would you please; in your own words. And please explain this to me from the Insurance companies point of view?

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From the customer's perspective: "My friend the probability geek tells me that insuring my house against natural isn't worth it, because the probability that I'll collect is only 2%, but the policy costs 3% of what it will pay me. In his words, the expected payoff is only half the cost of the policy. Big deal. The cost of the policy doesn't mean a lot to me -- I make enough money that, barring unforeseen catastrophes, the cost of insurance won't substantially affect my standard of living. But that unforeseen catastrophe worries me. IThe affect on my standard of living of the cost of the policy is less than 1% the affect of losing my house to flooding. So if I think in terms of standard of living, rather than money, it is worth it."

From the company's perspective: "The expected value is positive for us. Since we will insure many policies with the same terms, our long-term average payoff per policy will be approximately the expected payoff. So we use the expected payoff to decide that this is a good policy to offer."

Very good - you see the words as written and respond to them. Ignoring the actual context.
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Quote:
Originally Posted by SBMC
Oh…you would no want to do that

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Guess again. Once again, please let your interlocutors decide for themselves what they do and do not want to do.

You just did above…lets investigate again:

I posted:

…I roll a million times the averages work out - I know that - and that is what you all are talking about - but you hafve to roll many times for those probabilities to come about be them in your favor or not (as in, for an extreme example, the first 30,00 rolls are crits, the next are not, or vice versa). I'll still take the guarantee. It's like buying insurance; you give something up (money) but whenever you need it, no matter what, you know it will be there. Sure "maybe" I could hit it big in the stock market with that money and end up ahead in the long run...would you take that chance? Depends on your own desires…

Then you posted

Depends on my situation. The idea behind insurance is that the customer, while taking a policy with a negative expected value, will get the money in the contingency that he'll need it the most. A concave down utility function -- the same money is more important to him in adversity than in prosperity. I don't see that applying here.

I replied:

Really - explain further without regurgitating a textbook word for word for me would you please; in your own words. And please explain this to me from the Insurance companies point of view? Oh…you would no want to do that since it would support the point I was making which again, like many of my earlier statements you commented on in a similar fashion, was just an example to foster a comparison of perspective not mathematics - and unless you are completely daft you knew that.

Then you posted:

From the customer's perspective: "My friend the probability geek tells me that insuring my house against natural isn't worth it, because the probability that I'll collect is only 2%, but the policy costs 3% of what it will pay me. In his words, the expected payoff is only half the cost of the policy. Big deal. The cost of the policy doesn't mean a lot to me -- I make enough money that, barring unforeseen catastrophes, the cost of insurance won't substantially affect my standard of living. But that unforeseen catastrophe worries me. IThe affect on my standard of living of the cost of the policy is less than 1% the affect of losing my house to flooding. So if I think in terms of standard of living, rather than money, it is worth it."

From the company's perspective: "The expected value is positive for us. Since we will insure many policies with the same terms, our long-term average payoff per policy will be approximately the expected payoff. So we use the expected payoff to decide that this is a good policy to offer".

With the relevant part being:

From the company's perspective: "The expected value is positive for us. Since we will insure many policies with the same terms, our long-term average payoff per policy will be approximately the expected payoff. So we use the expected payoff to decide that this is a good policy to offer".

And the answer is - the opposite is true for the insurance company versus the insured! Good to see you were able to pull that point out of there.
Now…lets see I was trying to make a metaphor using a perspective of insurance (as it is familiar to many and thus, as metaphor's do in these circumstances aids in communicating) and my original point was…OH YA! That the +2 is a guarantee payoff every time…lets see the connection here…Improved Critical only has a chance to happen not a guarantee…so if I take Weapon Specialization I have a guaranteed payoff every time…do I really need to flow this into your posts? I think most can see this - if you can't then I would seriously consider getting a social life outside of the internet.

And again another example of attacking words and taking things out of context
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Quote:
Originally Posted by SBMC
since it would support the point I was making
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Unless you want to argue that the fighter's situation is analogous to the homeowner's, no, I don't see how it supports your point. I don't see making a single hit with neither a critical nor a damage bonus beyond your strength modifier is equivalent to losing your house to flooding.

I just addressed this above - and yes you did just support my point above.
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Quote:
Originally Posted by SBMC
"Come to naught" - then so would the Critical; a wash numericly.

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Which is why I began *my* calculations with a hit -- since neither feat under consideration affects the probability of getting that hit, we can ignore it. I was merely pointing out that your description of a bonus that affects every attack was inaccurate incontext. Tangential to the point of discussion, so I see no need to repeat it too often, but worth pointing it out once.

Ok I see…sure...I won't bother listing this one out there is plenty above and below.
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Quote:
Originally Posted by SBMC
No matter how you do it the numbers will, in the long run, represent a bell curve

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Which numbers do you have in mind? When you first mentioned a bell curve, I understood you were talking about attack rolls, and as I noted above, they have a uniform, not bell-shaped, distribution. It is completely unclear from the context which random variable you are referring to here. Please clarify if you wish me to decide whether I agree that the distribution is bell-shaped.

Please explain here why that is even important? It was a statement made in context with the rest of the surrounding words, which are, by the way (including the portion of the post it was in response to):

Quote:
Originally Posted by orsal
...that's so unlikely that you can reasonably ignore the possibility. If you want to say "but there is that nonzero probability, however small".. I'll point out that there is also a nonzero probability that in your 23,000 rolls you won't even get a *hit*, so the +2 damage would then come to naught.

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"Come to naught" - then so would the Critical; a wash numericly.

Unlikely? Your, and everyone elses, basis is on probabilities. This is no more probable than any other set combination of results that is similar. No matter how you do it the numbers will, in the long run, represent a bell curve and everyone here knows it - and we are all talking about actual hits not necessarily rolls of the die - again; attacking the words; for what purpose? How all that comes to be over time and iterations is pure chance - read game theory.

So lets see here - in English that would mean the context was to state, simply, over time random events create a bell curve…hmmm a tough concept to pull out of there.
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Are we done now Mr. Orsal or are you still all focused on tearing up words - are you a lawyer or law student by the way? Because lawyers are very good and yanking out irrelevant semantics and touting them as fact and things they are not; they are also very good at taking a particular piece of evidence (in this case words) and separating it from everything around it and beating it to death; since they can't do it with the support of the context; or at least it is much harder. As a matter of fact perhaps the evidence would be irrelevant otherwise.
I never once stated, and as a matter of act stated it was not, that my choice noted would be supported by numbers. I stated that I felt it was a "better choice" in my OPINION because of the guaranteed 2 points of damage versus the random effects of obtaining a critical hit.