Why the focus on criticals?

[frankthedm]

I'm honestly baffled here. I don't have the books yet, but from what I've seen here on EN world it seems like an amazing number of powers, feats, and magic items only kick in on criticals.

Which are only on a 20 and there in no "Improved Crit" until epic tier.

The heck?

because bonus damage on a crit mathematically works out to be about 1/10th of that damage/effect per hit, assuming rolls of 10 or less usually miss. Its a cute psychological trick, the player thinks he is getting significant bonus damage, but really is not getting that much in the long run. The impact of extra effects on the crit is even less since the real potency of 'effects' comes from applying them at opportune times rather than randomly slinging them onto a crit that may have killed the foe anyways.

 

[theNater]

You asked for it, you got it.

Assumptions: Using a single fair die, whether a d4, d6, d8, d10, or d12, we will roll it up to four times. We only take a second roll if the first was maximal, we only take a third roll if the second roll was taken and maximal, and we only take a fourth roll if the third roll was taken and maximal. Then we add together the results of all rolls we took and that is our value for the roll.

Procedures: We use the expected value formula to find the expected value of our rolls. The expected value is the sum of the products of the possible values with their probabilities. The formula as it applies to the d4 is in the spoiler block for those who are curious.
[sblock] (1*(1/4))+(2*(1/4))+(3*(1/4))+(4*0)+(5*(1/16))+(6*(1/16))+(7*(1/16))+(8*0)+(9*(1/64)+(10*(1/64))+(11*(1/64))+(12*0)+(13*(1/256))+(14*(1/256))+(15*(1/256))+(16*(1/256))
Note that 4, 8, and 12 do not contribute as they are totals we cannot possibly achieve by rolling a d4 in this manner.[/sblock]
Results: The approximate expected values of the various die types are as follows:

d4: 3.32
d6: 4.20
d8: 5.14
d10: 6.11
d12: 7.09

Conclusions: Under this rolling method, each time we step up the die type we increase the expected value by a little less than 1. The larger die types have higher expected values, but not massively higher.

 

[brehobit]

d4: 3.32
d6: 4.20
d8: 5.14
d10: 6.11
d12: 7.09

Conclusions: Under this rolling method, each time we step up the die type we increase the expected value by a little less than 1. The larger die types have higher expected values, but not massively higher.

Been done here before, but the formula for this kind of thing follows. X is the "sides of the die" and N is the expected value.

N=(X+1)/2+N/X
(Basic idea, on a given roll you get the normal value for the die: (X+1)/2. But one time in X you get to start over (so N/X).

Doing some math:

N-N/X=(X+1)/2
N*(1-1/X)=(X+1)/2
N*((X-1)/X)=(X+1)/2
N=X/(X-1)*(X+1)/2
N=X(X+1)/2(X-1)

Which yields about the same numbers you posted.

4 3.333333333
6 4.2
8 5.142857143
10 6.111111111
12 7.090909091
20 11.05263158


Mark

 

[theNater]

N=(X+1)/2+N/X
(Basic idea, on a given roll you get the normal value for the die: (X+1)/2. But one time in X you get to start over (so N/X).

Doing some math:

N-N/X=(X+1)/2
N*(1-1/X)=(X+1)/2
N*((X-1)/X)=(X+1)/2
N=X/(X-1)*(X+1)/2
N=X(X+1)/2(X-1)

Nice! Now I feel silly about doing it the long way.

I just don't think recursively, I guess.

 

[HP Dreadnought]

So how exactly does vorpal work?

 

[hong]

So how exactly does vorpal work?

When you roll maximum on a damage die, you reroll and add. Aka "exploding dice".

I assume that crits count as maximum rolls, so you can do lots of damage with a vorpal crit.

 

[Duelpersonality]

So how exactly does vorpal work?

Any time you roll the maximum result on a damage die, you get to roll another damage die until you stop rolling max damage.

Results: The approximate expected values of the various die types are as follows:

d4: 3.32

Would the falchion and glaive just deal double that amount?

 

[Henry]

I'm not sure I would assume only 10 attack rolls per encounter. Assuming most combats will likely last for 5 rounds to 10 rounds, and there are 5 people in wotc's assumed average group, you're talking about somewhere between 25 and 50 (!) attack rolls per encounter, not even counting the DM. That's at least twice you'll see a crit on the table in an encounter, possibly three or four.

 

[Storm-Bringer]

You asked for it, you got it.

Assumptions: Using a single fair die, whether a d4, d6, d8, d10, or d12, we will roll it up to four times. We only take a second roll if the first was maximal, we only take a third roll if the second roll was taken and maximal, and we only take a fourth roll if the third roll was taken and maximal. Then we add together the results of all rolls we took and that is our value for the roll.

Procedures: We use the expected value formula to find the expected value of our rolls. The expected value is the sum of the products of the possible values with their probabilities. The formula as it applies to the d4 is in the spoiler block for those who are curious.
[sblock] (1*(1/4))+(2*(1/4))+(3*(1/4))+(4*0)+(5*(1/16))+(6*(1/16))+(7*(1/16))+(8*0)+(9*(1/64)+(10*(1/64))+(11*(1/64))+(12*0)+(13*(1/256))+(14*(1/256))+(15*(1/256))+(16*(1/256))
Note that 4, 8, and 12 do not contribute as they are totals we cannot possibly achieve by rolling a d4 in this manner.[/sblock]
Results: The approximate expected values of the various die types are as follows:

d4: 3.32
d6: 4.20
d8: 5.14
d10: 6.11
d12: 7.09

Conclusions: Under this rolling method, each time we step up the die type we increase the expected value by a little less than 1. The larger die types have higher expected values, but not massively higher.

So, what were those crazy numbers I was coming up with?

 

[Celebrim]

I'm not sure I would assume only 10 attack rolls per encounter. Assuming most combats will likely last for 5 rounds to 10 rounds, and there are 5 people in wotc's assumed average group, you're talking about somewhere between 25 and 50 (!) attack rolls per encounter, not even counting the DM. That's at least twice you'll see a crit on the table in an encounter, possibly three or four.

Yes, but that's really no incentive to take bonus damage on a critical if you have alternatives.

Yes, you might do 3d8 extra damage on 5 attacks, but if you had the option to instead do +2 damage on all 50 attacks, that would be even better. In fact, you don't start coming out ahead compared to +2 damage until something around +5d8 bonus damage per critical.

Things that inflict conditions on the target on a critical are probably more worth it, especially against solo monsters, but you have to figure out how valuable exploiting that condition is. Things also get much better with improved criticals. But, that's what retraining is for.