Halfling luck, advantage and improved critical oh my!

[Spacehamster]

So got a question for those that feel a bit mathematical.
what is the chance to score a critical strike with halfling luck, advantage and 19-20 crit modifier? :)

[Yorrin]

The fact that my numbers came out so linear makes me feel like I did it wrong, but here's my quick attempt:

19-20 gives you a 10% base. adv bumps that up to 20%, and lucky bumps that up to 30%

edit: no, I definitely mis-calculated halfling luck. recalculating...

lucky adds like 1%? roughly? I'll wait for people who are more current with tier math.

[JellyPooga]

19-20 gives you a 10% chance of a crit.

Halfling Lucky means you get to re-roll on a 1 (5% chance)

That re-roll then has its own 10% chance of a crit, so 5% then 10% is an additional 0.5% chance.

So you're looking at a 10.5% chance, assuming you don't have advantage (with advantage, it's 21%)

[Bladeyeoman]

So you're looking at a 10.5% chance, assuming you don't have advantage (with advantage, it's 21%)

The thing to remember is that advantage doesn't quite double your chance to crit, since when both dice crit, that doesn't get to count as two crits.

So, as described above, on a single roll there's a 10.5% chance of criting. To look at advantage, counterintuitively, it's easier to calculate the probability of NOT criting, and then work from there.

Per roll there's a 10.5% chance of critting, so an 89.5% chance of not criting. What are the odds that both rolls are not crits? 89.5% * 89.5%, or 0.895^2, which is 0.801025. Since you either crit or don't crit with probability 1, the probability of you criting is 1 minus the probability of you NOT criting. Which is 1-0.801, or 0.199.

tl;dr: 19.9% chance of criting with advantage and halfling luck.

Sidenote: this is slightly lower than if we had added the two independent probabilities (10.5%) together. Which makes sense, because as mentioned at the top, we have to account for the fact that on rare occasions both dice will crit, but we only get to count one of them.

[Aetol]

Don't forget that when you have advantage, if both dice get a crit it only counts as one crit.

The easiest way is to calculate the odds of not having a crit on two rolls : 89.5%^2 = 80.1% (rounded). So you have a 19.9% chance of getting a crit.

EDIT : ninja'd.

[Bladeyeoman]

Sidenote: using the same approach, we can find that your probability for criting with improved crit but WITHOUT halfling luck is 19%.

Bonus sidenote: Probabilities are fun!

The probability of event A occurring or event B occurring (which we can write P(A OR B) ) is equal to the probability of A plus the probability of B MINUS THE PROBABILITY OF A AND B. So

P(A OR B) = P(A) + P(B) - P(A and B)

So, we could also calculate our chance of crit with advantage as the probability of rolling a crit on one die (P(A)=10%) plus the probability of rolling a crit on the other die (P(B)=10%) minus the probability of rolling a crit on both dice at the same time (P(A and B)=1%). Which gives us 19%! Same as the other approach!

The formal part of the other approach:

The probability of event A happening and the probability of event A not happening sum to 1.
P(A) + P(!A)=1

So if we can calculate the probability of event A not happening ( P(!A) ) it is trivial to calculate the probability of event A happening. And for event A (where A is "we rolled a crit!") to NOT occur, we must roll a 1-18 on both dice.

So now we're looking at P(B and C) where B is "first die was a 1-18" and C is "second die was a 1-18". In probability, for independent events like two die rolls, "and" is easy to caculate: P(B and C) = P(B) * P(C)

[Bladeyeoman]

Oop! Page 173 of the PHB tells us that halfling luck and other things that let you reroll a dice on an attack roll can only be applied to one of the dice. That said, this is relevant only in the event that we roll two 1s, which will occur 0.25% of the time. So the above estimates are very close to being correct.

UPDATE:
Mapped out all possible combinations of die rolls, and the probability of getting a crit with advantage and halfling luck (when we correctly apply it) is 0.19925.

[bid]

Lucky has very little impact:
- 1d20 = 5% crit
- 2d20 = 9.75% crit
- 3d20 = 10.2125% crit

If you look at the 20*20 cases for which lucky activates:
- 1 case where first was 20 and second was 1 (already a crit before lucky)
- 1 case where first was 1 and second was 20 (already a crit before lucky)
- 1 case where both roll 1s (lucky!)
- 18 cases where first rolled 1 and second rolled 2-19 (lucky!)
- 18 cases where first rolled 2-19 and second rolled 1 (lucky!)

So we have {18 + 18 + 1 = 37}/400 of rerolling and 1/20 of critting that reroll, or 37/8000 of lucky-critting.


With champion 19-20 it's 35/8000 * 2 (since the 19:1 pairs are already crits):
- 1d20 = 10% crit
- 2d20 = 19% crit
- 3d20 = 19.875% crit

EDIT - might as well get to level 15...

With champion 18-20 it's 33/8000 * 3 (since the 18:1 pairs are already crits):
- 1d20 = 15% crit
- 2d20 = 27.75% crit
- 3d20 = 28.9875% crit

EDIT2 - forgot to double/triple the lucky crit. Now it matches Aetol.

[Aetol]

UPDATE:
Mapped out all possible combinations of die rolls, and the probability of getting a crit with advantage and halfling luck (when we correctly apply it) is 0.19925.

Weird, it's not what I find by doing the math.


Probability of getting a crit without a reroll : 1 - 0.9² = 0.19
Probability of not getting a crit without a reroll : 0.9² = 0.81
(Probability of then getting a 1 on one of the dice : 1/18)
Probability of then getting a 1 on any of the dice : 1 - (17/18)² = 35/324
Probability of then getting a crit by rerolling that 1 : 0.1
Hence, probability of getting a crit by getting a 1 and rerolling it : 0.81*(35/324)*0.1 = 7/800 = 0.00875

So the total probability of getting a crit is 0.19875, or 19.875%.

There's 0.5% missing. Did I do something wrong ?

[JellyPooga]

Minutiae are fun, for sure, but it's probably more useful to ballpark it at roughly 1-in-5 :smallwink: