Alrighty, bored at work so I just worked out a probability table of "roll twice, take the higher result". For a d4, I got that the average increases by .75, bringing it to a nice round 3.25 average. For a d20, I got an average increase by ~3.25, giving us an average of about 13.75.
Then I remembered I've never taken a statistics class and probably got this entirely wrong.
Anyone have any real math on "roll twice, take the higher" on a d20?
should give the total of all the roll/reroll results for a d20. that is: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 (possible results if 'reroll' is 1) 2,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 (possible results if 'reroll' is 2) 3,3,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 (possible results if 'reroll' is 3) etc...
It involves value frequency. For example, to get a value of 1, there's one combination. To get a value of 2, there's three combinations (1,2; 2,1; 2,2), to get 3, there's five sets (1,3; 2,3; 3,3; 3,2; 3,1), to get 4 there's 7 sets, and so on.
There are 2 more ways to get any progressively higher number. 1 way to get 1. 3 ways to get 2. 5 ways to get 3. 7 ways to get 4. 9 ways to get 5.
There are 2X-1 ways to get X.
There's a simple and elegant way to turn this into a formula, but I'm at work and can't be bothered. Excel ahoy!
Summation of all values is 5530. 5530/400 = 13.825
Further details: 400 lines. First column is a repeated series from 1 to 20. Second column is a series of twenty 1s, twenty 2s, and so on, until we get twenty 20s. Third column is Max of column 1 and 2. Summation of column three is 5530. 5530/400 = 13.825. Compared to the average of a single d20 (10.5), it's +3.325.
(This, btw, is why Danger Sense provides less of a benefit than Improved Initiative in general, but makes a difference depending on the "DC" you're trying to hit.)
Roll 2 D20s and take the higher: Mean:13.825 Median: 15
Normal D20 roll has a mean of 10.5. This may sound like a general increase of 3.325, but that is not a good way of looking at it.
About 50% of all rolls on a single d20 will range from 5 to 16.
About 50% of all rolls taking the higher of 2 d20s will range 10 to 18.
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About 75% of all rolls on a single d20 will range on average 3.5 to 17.5
About 75% of all rolls taking the higher of 2 d20s will range from 8 to 19
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About 90% of all rolls on a single d20 will range from 2 to 19
About 90% of all rolls taking the higher of 2 d20s will range from 5 to 20.
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About 95% of all rolls on a single d20 will range on average 1.5 to 19.5
About 95% of all rolls taking the higher of 2 d20s will range from 4 to 20.
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50% of the time with a single D20 roll, you will roll under average (10.5).
42% of the time with taking the higher of 2 d20s, you will roll under average(13.825).
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50% of the time with a single D20 roll, you will roll 10 and under.
25% of the time with taking the higher of 2 d20s, you will roll 10 and under.
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Needing a 20 for a Critical Hit
A single d20 roll has a 5% chance to score a critical hit
Taking the higher of 2 d20s has a 9.75% chance to score a critical hit.
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Needing a 19 or 20 for a Critical Hit. (Weapon Mastery or Jagged Weapon)
A single d20 roll has a 10% chance to score a critical hit.
Taking the higher of 2 d20s has a 19% chance to score a critical hit.
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Needing an 18 or higher for a Critical Hit. (Daggermaster or Student of Caiphon(with Radiant keyword) PPs.)
A single d20 roll has a 15% chance to score a critical hit.
Taking the higher of 2 d20s has a 27.75% chance of scoring a critical hit.
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Needing a 17 or higher for a Critical Hit. (Raise the Stakes - Rogue Daily Utility Level 16)
A single d20 roll has a 20% chance to score a critical hit.
Taking the higher of 2 d20s has a 36% chance to score a critical hit.
Starting to see the boon? An Avenger/Rogue wielding going Daggermaster and wielding daggers or an Avenger/Warlock wielding a Radiant(keyword) Weapon going Student of Caiphon will both have 27.75% chance to Crit, and will never roll under 10 75% of the time.
Now excuse me while I start setting up a Half-elf Wis/Str Avenger, with Dilettante ability being Twin Strike, taking the Pact Initiate feat(Star), and going for the Student of Caiphon paragon path with the plan on wielding two Radiant Katars. And the Punisher of the Gods Epic Destiny too. Talk about broken, 48% chance to roll at least one critical hit against a foe using Twin Strike against. ~7.7% chance for two critical hits in that 48% chance.
Seems to me that the same half-elf (twin strike) Avenger would be better served (damage-wise) by going Sneak of Shadows -> Daggermaster with a pair of bloodiron daggers. Sure, the katar is high crit, but the dagger gets the 18-20 crit chance without the enchantment, and the bloodiron is 1d10 (and again at the beginning of the next turn, so effectively 2d10 per plus) rather than the 1d6 from radiant.
On a crit at level 14 (first level you can get a Radiant weapon off creation, and that's only one), the Student would do 6 (base)+ 2d6 (high crit)+ 3d6 (crit effect) +3 (weapon plus) + 3 (property) = 29.5 expected damage from the weapon, with a minimum of 17 and a maximum of 42.
At the same level (14, and you can have TWO bloodiron daggers) the Daggermaster would do 4 (base) + 3d10(crit effect) + 3(weapon plus) = 23.5 expected damage on that turn, with another 16.5 next turn for 40 expected weapon damage; minimum damage on that is 13 and a maximum of 67.
So I guess it depends on whether you're risk-averse or not. The Student/Radiant Katar option has a higher average damage per crit, but bloodiron offers more upside (and potential risk, as well). Unless I've botched the calculations somewhere.
(Disclaimer: calculations are for expected damage on a crit, without to-hit numbers factored in because they use the same 1[w] attack power using the same stat and weapons with the same proficiency bonus. DPR calculations will return lower numbers, and may push the advantage back over to the Student/Radiant Katar route. I may work those out when I get some more time.).
Now excuse me while I start setting up a Half-elf Wis/Str Avenger, with Dilettante ability being Twin Strike, taking the Pact Initiate feat(Star), and going for the Student of Caiphon paragon path with the plan on wielding two Radiant Katars. And the Punisher of the Gods Epic Destiny too. Talk about broken, 48% chance to roll at least one critical hit against a foe using Twin Strike against. ~7.7% chance for two critical hits in that 48% chance.
Unveiled Visage/Punisher of the Gods creates a nasty/ridiculous combo. Unveiled Visage, an Avenger PP, gives you back a Channel Divinity, RRoT, when you spend an AP. So as long as you hit with a RRoT attack, you crit and generate AP until you run out of actions for the round. Hitting with an Armor Splinter/Triumphant Attack (a very good chance) imposes a penalty that means you're gonna hit anything from then on unless you roll double 1's, even with Power Attack. Armor Splinter has two attacks, which gives you 4d20. Really just one of those attacks needs to hit to impose a hefty penalty. With Martial Mastery (reloading and using Armor Splinter every round) this a continual cycle until you miss with Armor Splinter. Toss in Bloodiron/Rending, Two Weapon Opening, and other stuff that's good, and it quickly starts to be stupid.
I think it should be pointed out that the average value for rolling d20 twice is almost never useful information. The value that shows on a d20 isn't a scaling factor relative to its worth. Example: you are trying to hit something with a defense of 20, and you have +10 to your attack roll:
Your "average attack roll" is going to be 10+13.825, or 23.8, compared to 20.5 rolling once. So you might be tempted to think that rolling twice improves your chance to hit by 15% (i.e. by 3 numbers on the die).
In actuality though, averaging doesn't provide meaningful information, because if you need a 10 to hit, a 1 is exactly as valuable as a 9, and a 10 is exactly as valuable as a 19. The number of times you can roll a d20 twice and not get at least one number higher than a 9 is .45^2 or 20.25% of the time. So you will hit the target's defense 79.75% of the time, compared to 55% of the time with only one die.
That means 24.75% of the time you will hit because of rolling a second die. Compare that to the 15% value you'd expect if you averaged them.
