D20 Math Question

[Crow]

I need a little help deciding upon which way to go on something.

In a D20 system, If you make three attacks during a round, which of these options will net you more successful hits on average?

1. A flat +2 to hit on all attacks

2. The ability to reroll one of those attacks

Just eyeballing it, it seems like the flat +2 is better, but I was never any good at math.

[AvatarVecna]

It depends on what you need to succeed normally, and also on whether you're stuck with the reroll or if you take the better of the two rolls.

EDIT: Assuming that you take the better of two rolls with the reroll (essentially being like 5e advantage), the +2 to all attacks would be better unless the roll you need is particularly low.

[TheifofZ]

Depends on where those attacks are coming from. If it's from BAB progression, then that means 0/-5/-10. If it's from Multi-attack, that's 0-2/0-2/0-2.
If we assume the target AC is 10, then the base rolls to hit are: 10/15/20 or 12/12/12.
That is: 50%/25%/5%, or 40%/40%/40%.
Adding a +2 across the board changes it to 8 (60%)/13 (35%)/18 (15%), and 10 (50%)/10/10.
... I don't do enough math to be able to tell you the actual statistical advantage of rerolls, especially to BaB progression, though it doesn't seem like it would be better.

[AvatarVecna]

Ultimately, it all depends on two more things:

1. What the target number is.

2. How the reroll works (is it 'roll twice, take the better roll', 'reroll if you don't like the first roll, but you're stuck with the second', or what?)

EDIT: Okay, I ran the numbers, assuming it works like 5e advantage. The +2 is always better, unless you can apply the reroll to multiple attacks.

[Sacrieur]

In the best case scenario, rerolling just one attack is disadvantageous, since it has an average of 11.4 across three attacks and +2 has an average of 12.5.

[Crow]

Ultimately, it all depends on two more things:

1. What the target number is.

2. How the reroll works (is it 'roll twice, take the better roll', 'reroll if you don't like the first roll, but you're stuck with the second', or what?)

EDIT: Okay, I ran the numbers, assuming it works like 5e advantage. The +2 is always better, unless you can apply the reroll to multiple attacks.

Sorry for the lack of info, doh.

Attack is not BAB progression, but multiattack style

Target number is generally between 8-12, with outliers of 4 and 16

Reroll is "stuck with the new one"

[Bakkan]

One more relevant question: do we know whether or not an attack succeeded before deciding whether or not to reroll?

[Edit] If not, can we use information from earlier rolls to decide whether to reroll on later ones, or do we have to make all rolls before finding out about the success or failure of any if them? For example, if my first roll is a 12 and misses, can I use that information to decide to reroll my second roll if it's a 12 or less?

[Sacrieur]

One more relevant question: do we know whether or not an attack succeeded before deciding whether or not to reroll?

It's not really relevant, since it doesn't change the fact one of your numbers is re-rolled. The average for 2d20, dropping the lowest, is 13.33.

[AvatarVecna]

It's not really relevant, since it doesn't change the fact one of your numbers is re-rolled. The average for 2d20, dropping the lowest, is 13.33.

The problem here is that the OP has said that it's not "2d20b1", it's "if you don't like your roll, roll again, but you're stuck with the second one. So if you rolled, say, a 3, and decided to take the reroll and got a 1, you'd be stuck with the 1, even though the 3 was better.

[Sacrieur]

The problem here is that the OP has said that it's not "2d20b1", it's "if you don't like your roll, roll again, but you're stuck with the second one. So if you rolled, say, a 3, and decided to take the reroll and got a 1, you'd be stuck with the 1, even though the 3 was better.

Right, so if we take the best case scenario (2d20d1) and if it's still not good enough, then any other methods worse than that are similarly going to be no good.

Also I built a program to test this out and got an average value of 13.8 for 2d20d1

[Jack_Simth]

It depends on what you need to succeed normally, and also on whether you're stuck with the reroll or if you take the better of the two rolls.

EDIT: Assuming that you take the better of two rolls with the reroll (essentially being like 5e advantage), the +2 to all attacks would be better unless the roll you need is particularly low.
Or particularly high. If you only miss on a 1 anyway, the +2 doesn't change the odds, the reroll does (reduces the odds of 1's slightly). Likewise, if you only hit on a 20 even with the +2, then the +2 doesn't help (by definition), while the reroll does (increases the odds of 20's slightly).

Outside of those outlying situations, however, the +2 to all is going to be strictly better than reroll on one of the dice. In those outlying situations, the difference is likely not significant anyway, because the encounter is either hard enough that you'll get slaughtered anyway (need a 20 even with the +2), or so easy it's largely irrelevant (only miss on a 1, even without the +2).

So yeah, the +2 is better.

[Sacrieur]

Or particularly high. If you only miss on a 1 anyway, the +2 doesn't change the odds, the reroll does (reduces the odds of 1's slightly). Likewise, if you only hit on a 20 even with the +2, then the +2 doesn't help (by definition), while the reroll does (increases the odds of 20's slightly).

It doubles the chances of rolling any number. That's far from a slight increase.

But it halves the chance of rolling a 1. So in other words you have a 10% chance to roll a 20, but only a 2.5% chance of ending up with two 1's. I didn't factor this into my program.



Outside of those outlying situations, however, the +2 to all is going to be strictly better than reroll on one of the dice. In those outlying situations, the difference is likely not significant anyway, because the encounter is either hard enough that you'll get slaughtered anyway (need a 20 even with the +2), or so easy it's largely irrelevant (only miss on a 1, even without the +2).

So yeah, the +2 is better.

With one attack, a +2 yields an average value of 12.5, while 2d20d1 has an average of 13.8.

[Jack_Simth]

It doubles the chances of rolling any number. That's far from a slight increase.

Ehhh... no. That depends on the reroll logic you use. If you reroll anything except the target number, this is almost true. However: You're only starting with a 5%, so the increase while almost double, is still slight compared to 100%. Take the 'need a 20' scenario:

You have a 5% chance of rolling a 20 straight up, and not bothering with the reroll on this attack.
For the other 95%, you have a 5% chance of rolling a 20. That's not quite 10%. That's 5%, + 95% of 5% = 5% + 4.75% = 9.75%

If you're changing a 5% chance of a hit to a 9.75% chance of a hit, you've nearly doubled your hit rate. But it's not going to matter, as if you actually need a 20 to hit it, the critter is liable to steamroll you anyway, so changing the hit rate by almost 1 in 20 is largely irrelevant.



But it halves the chance of rolling a 1. So in other words you have a 10% chance to roll a 20, but only a 2.5% chance of ending up with two 1's. I didn't factor this into my program.
This is also incorrect. It doesn't halve the odds of rolling a 1. Take the scenario where you only reroll 1's, as all you need is a 2:

You have a 95% chance of not needing the reroll, a 5% chance of rolling a 1 on the first throw.
If you do roll a 1, then you have a 95% chance of getting something that's not a 1, and a 5% chance of getting something that is a one. So the chance of getting a 1 is 5% of 5%: One quarter of 1%, or 0.25%. One in 400 vs. one in 20.


With one attack, a +2 yields an average value of 12.5, while 2d20d1 has an average of 13.8.2d20 drop lowest 1 averages 13.825, specifically (there's only 400 combinations on 2d20, so it's easy to brute-force with a simple spreadsheet). However: The scenario was +2 on three attacks vs. reroll one of those three attacks. So while yes, if you know the target value, the reroll is more useful on a single attack... across three, the +2 is more valuable (other than the edge cases where the +2 doesn't matter, either because you need a 20 even with the +2, or you only fail on a 1 even without the +2).

Or to put it another way: Which is greater: 13.825 + 10.5 + 10.5, or 12.5*3?

[Hamste]

It doubles the chances of rolling any number. That's far from a slight increase.

But it halves the chance of rolling a 1. So in other words you have a 10% chance to roll a 20, but only a 2.5% chance of ending up with two 1's. I didn't factor this into my program.




With one attack, a +2 yields an average value of 12.5, while 2d20d1 has an average of 13.8.

You actually have a 1 in 400 chance or .25% chance of rolling a 1 with 2d20 take best. 400 possibilities only one of which is 1,1. Similarly there is 39 possibilities out of 400 that get you 20 so it is not a 10% chance to roll a 20 but a 9.5% chance.


http://anydice.com/program/b38

Here is the answers from a dice roller.

Of course this falls apart if you don't roll 2d20b1 but instead choose whether to reroll or not. Assuming you reroll when 10 or less (as that is below average) you have the normal chance of 11-20 and a 50% chance to reroll. Rerolls would then have normal chances. So 11-20 would have 75% chance (50% that first time and 50% the second) and 1-10 would have 25%. You would have a 75% chance to roll 11 or above and that would be divided evenly among 11-20. So there is 7.5% chance of getting a 20 while there would be only a 2.5% chance of a 1. Average is 13.333 I think but the math might be off a bit. For any value you wish you just need to figure out the percent chance you roll below your cut off point and square it, that is the chance you roll below your cut off point and 1-the cut off point is the chance to roll above that.

[meschlum]

Rerolls come in lots of flavors, depending on what you know at the time and what you get to keep.

I'll assume you know the target AC, which is helpful for rerolls since you know whether you want to reroll or not. Since you know whether your first roll hits, it's the same as keeping the best of two rolls, as a successful first roll is not rerolled, and a failed first roll is rerolled, and if the second passes, you get the best.

If you have a probability of P of hitting (and know it), then you have a probability of 1 - (1 - P)^2 of hitting with a single reroll on one die - that's 2 * P - P^2, so a bit less than twice P when the probability is low.

Since you're rolling three attacks, you can benefit from your reroll even if your first attacks hit. So in all, you get an average of 4 * P - P^4 hits instead.

As a secondary assumption, I'll assume there are no modifiers to the base roll, to make things simpler. You can offset the values by the roll bonus otherwise (a base +10 ot the roll means the first line is at 32+ instead, for instance).



AC 22: need a 20 to hit, even with +2. Reroll is better.
AC 21: need a 20 to hit with reroll, 19 to hit without. With one reroll, you have a 9.999375% chance of hitting, and a 10% chance with +2. Since the reroll is for one die only, it's either 3 attacks * 10% = average of 0.3 hits, or 0.19999375 hits with a reroll. +2 is better.
AC 20: need 20 / 18. Reroll is still 9.999375% chance of hitting, +2 is 15%. +2 is better (0.45 hits to 0.1975).
AC 19: need 19 / 17. Reroll is 19.99%, for a total of 0.3999 hits on average, or 0.6 with +2.

For ACs down to 4, the +2 gives you 3 * (P + 0.1), so the difference is P - P^4 - 0.3. The maximum of P - P^4 is 0.47 when P is close to 0.63, so a reroll can be better.

In fact, we find that for P between 0.35 and 0.85 (inclusive, so AC 14 to 4), the reroll gives superior results, though the increase isn't much (0.17 hits at most).

For AC 3, need 3 / 2. Reroll gives 2.94 hits. With the +2, it's 0.95 * 3 = 2.85 hits, reroll is still better.

For AC 2 or less, need a 2 in every case, so the reroll is better since you roll 4 times rather than 3.


So the +2 bonus is better for high AC (15 or more), when you know after each roll whether it would hit or not and you can assign the reroll freely (rather than only getting to reroll your first attack if it misses). The logic is that when you're reasonably likely to hit anyway, a reroll will compensate for poor luck where a +2 won't really increase your odds much.

The benefits of a single reroll are reduced by having more attacks, though it takes 5 or more attacks for AC 4 to be better with a +2. High ACs benefit from a +2 much faster (at 4 attacks, a +2 is better from AC 13 and more).

This is all based on wanting to get the average number of hits, not estimating the probability of getting a single hit or some such.


And if you don't know whether you should reroll, things get complicated (especially since you learn about the opponent's AC as you attack - if your first roll of 12 missed, you know you want to reroll any 12 or less you roll later on).


An issue with some of the math in previous posts is that if you hit you don't need to reroll the die, so it's not one (possibly) rerolled die and two others.


In depth reroll analysis for 3 attacks:

With probability P^3, you hit 3 times, no rerolls needed
With probability 3 P^2 (1 - P), you hit twice, and miss once, so the reroll means you hit 2 + P times on average (instead of 2).
With probability 3 P (1 - P)^2, you hit once, and miss twice, benefitting from the reroll once for 1 + P hits on average.
With probability (1 - P)^3 you miss three times, the single reroll means you have an average of P hits.

So you get 3 P hits from the baseline, and (1 - P^3) * P hits from the rerolls, for a total of 4 P - P^4 hits on average.

[Jack_Simth]

Of course this falls apart if you don't roll 2d20b1 but instead choose whether to reroll or not. Assuming you reroll when 10 or less (as that is below average) you have the normal chance of 11-20 and a 50% chance to reroll. Rerolls would then have normal chances. So 11-20 would have 75% chance (50% that first time and 50% the second) and 1-10 would have 25%. You would have a 75% chance to roll 11 or above and that would be divided evenly among 11-20. So there is 7.5% chance of getting a 20 while there would be only a 2.5% chance of a 1. Average is 13.333 I think but the math might be off a bit. For any value you wish you just need to figure out the percent chance you roll below your cut off point and square it, that is the chance you roll below your cut off point and 1-the cut off point is the chance to roll above that.
Where it really gets complicated, though, is sorting out which of the die rolls you end up using the reroll on. Take the example of rerolling 10's or below. You have a 50% chance of eating up the reroll if you still have it when you get to that die roll.

So you've got a 50% chance of using it on the 1st, a 25% chance of using it on the second, a 12.5% chance of using it on the third, and a 12.5% chance of not using it at all.

So for that 1st roll, you've got 7.5% chance for each number on 11-20, and a 2.5% chance for each number on 1-10. Weighted average works out to 13 - even, apparently, unless I'm doing my spreadsheet wrong. Huh.

For the 2nd roll, you've got a 50% chance of having to abide by the 1st roll regardless, and a 50% chance of having your reroll available. So you've got a 6.125% chance for each number in the 11-20 range, and a 3.875% chance of each number in the 1-10 range. Weighted average works out to 11.625 (again, unless I did the math wrong somewhere).

For the 3rd roll, you've got a 75% chance of having to abide by the 1st roll regardless, and a 50% chance of having your reroll available. So you've got a 5.5625% chance for each number in the 11-20 range, and a 4.4375% chance for each number in the 1-10 range. Weighted average works out to 11.0625

So our average average is 11.895833333...

Vs. the average of the flat +2 as 12.5.

... assuming I did my math right. I may not have.

[Sacrieur]

Ehhh... no. That depends on the reroll logic you use. If you reroll anything except the target number, this is almost true. However: You're only starting with a 5%, so the increase while almost double, is still slight compared to 100%. Take the 'need a 20' scenario:

You have a 5% chance of rolling a 20 straight up, and not bothering with the reroll on this attack.
For the other 95%, you have a 5% chance of rolling a 20. That's not quite 10%. That's 5%, + 95% of 5% = 5% + 4.75% = 9.75%

I used the possibility of getting two 20s.



If you're changing a 5% chance of a hit to a 9.75% chance of a hit, you've nearly doubled your hit rate. But it's not going to matter, as if you actually need a 20 to hit it, the critter is liable to steamroll you anyway, so changing the hit rate by almost 1 in 20 is largely irrelevant.

I don't think so. I think it's a worthwhile investment. With three attacks you have a 15% chance that at least one of them will turn up as a 20, while a reroll increases those odds to 19.75%. So you go from 1 in 7 (ish) to 1 in 5 (less ish).



This is also incorrect. It doesn't halve the odds of rolling a 1. Take the scenario where you only reroll 1's, as all you need is a 2:

You have a 95% chance of not needing the reroll, a 5% chance of rolling a 1 on the first throw.
If you do roll a 1, then you have a 95% chance of getting something that's not a 1, and a 5% chance of getting something that is a one. So the chance of getting a 1 is 5% of 5%: One quarter of 1%, or 0.25%. One in 400 vs. one in 20.


You actually have a 1 in 400 chance or .25% chance of rolling a 1 with 2d20 take best. 400 possibilities only one of which is 1,1. Similarly there is 39 possibilities out of 400 that get you 20 so it is not a 10% chance to roll a 20 but a 9.5% chance.

Yeah, even my program shows 0.25%. I left out a zero.

[AzraelX]

Similarly there is 39 possibilities out of 400 that get you 20 so it is not a 10% chance to roll a 20 but a 9.5% chance.

39 / 400 = 9.75% chance.

[meschlum]

Where it really gets complicated, though, is sorting out which of the die rolls you end up using the reroll on. Take the example of rerolling 10's or below. You have a 50% chance of eating up the reroll if you still have it when you get to that die roll.

So you've got a 50% chance of using it on the 1st, a 25% chance of using it on the second, a 12.5% chance of using it on the third, and a 12.5% chance of not using it at all.

So for that 1st roll, you've got 7.5% chance for each number on 11-20, and a 2.5% chance for each number on 1-10. Weighted average works out to 13 - even, apparently, unless I'm doing my spreadsheet wrong. Huh.

For the 2nd roll, you've got a 50% chance of having to abide by the 1st roll regardless, and a 50% chance of having your reroll available. So you've got a 6.125% chance for each number in the 11-20 range, and a 3.875% chance of each number in the 1-10 range. Weighted average works out to 11.625 (again, unless I did the math wrong somewhere).

For the 3rd roll, you've got a 75% chance of having to abide by the 1st roll regardless, and a 50% chance of having your reroll available. So you've got a 5.5625% chance for each number in the 11-20 range, and a 4.4375% chance for each number in the 1-10 range. Weighted average works out to 11.0625

So our average average is 11.895833333...

Vs. the average of the flat +2 as 12.5.

... assuming I did my math right. I may not have.

The third roll is 75% and 25%, but that's a typo.

As a shortcut in the above case:

1-20, average is 10.5 ((1 + 20) / 2)
11-20, average is 15.5 ((11 + 20) / 2)
1-10, average is 5.5 ((1 + 10) / 2)

With a reroll at 10 or less, you have 15.5 half the time (no reroll) and 10.5 half the time (reroll), so an average of 13 overall.

1/8 you never reroll because all results were in the 11-20 range, average 15.5 * 3 = 46.5
3/8 you reroll once (10.5) and have two results in the 11-20 range, so 15.5 *2 + 10.5 = 41.5
3/8 you reroll once (10.5) and have one result in the 11-20 range and one in the 1-10 range. 31.5 total.
1/8 you reroll once (10.5) and have two results in the 1-10 range. 21.5 total

That gives a total of 287 / 8, so an average of 35.875 for three dice. The average for one die is one third of that, so 11.958333..., as stated.


Also, adding the average rolls doesn't tell you whether you actually hit or not. If the AC is 11, your average number of hits when rolling 10 or less is 0, not 5.5.


Another part is that in this case you learn about the opponent's AC as you roll... which is why I assume you know it, or things get messy quickly.

[Kurald Galain]

It doubles the chances of rolling any number.

Bear in mind that while "doubling" makes the result sound big, doubling a small number will in fact only give you a slightly less small number, not a big one.

And, of course, doubling a zero gives you back the same zero.

[meschlum]

I don't think so. I think it's a worthwhile investment. With three attacks you have a 15% chance that at least one of them will turn up as a 20, while a reroll increases those odds to 19.75%. So you go from 1 in 7 (ish) to 1 in 5 (less ish).


Yes, the reroll option is worth it at AC 22+ (always need a 20 even with +2), and for lower ACs (per my math above). However, to be picky, the probability of getting at least one 20 on 3 dice is not 15% - otherwise you'd be sure to get a 20 when rolling 20 dice!

It's 14.2625%, even closer to 1/7 (14.285614...%), as it's 1 - (19/20)^3. You get 0.15 hits on average because some of the time you roll more than one 20.


Time for a general rule!

If you roll A attacks, each with probability H of hitting, and have access to 1 reroll, then your average number of hits is equivalent to a bonus of B on each attack.

The formula is: B = (H - H ^ (A + 1)) / A

For instance, with 3 attacks and a 20% chance of hitting (17 or more), B is ~6.6%, so a reroll is better than a +1 and worse than a +2.

With 3 attacks and a 50% chance of hitting (11 or more), B is ~14.6%, so a reroll is better than a +2 and almost as good as a +3.


The formula does not work when you get into the sure hit / sure miss range (H + B > 0.95 or < 0.05), due to auto-hit and miss rules. Still useful, though!

[Sacrieur]

Yes, the reroll option is worth it at AC 22+ (always need a 20 even with +2), and for lower ACs (per my math above). However, to be picky, the probability of getting at least one 20 on 3 dice is not 15% - otherwise you'd be sure to get a 20 when rolling 20 dice!

It's 14.2625%, even closer to 1/7 (14.285614...%), as it's 1 - (19/20)^3. You get 0.15 hits on average because some of the time you roll more than one 20.


Time for a general rule!

If you roll A attacks, each with probability H of hitting, and have access to 1 reroll, then your average number of hits is equivalent to a bonus of B on each attack.

The formula is: B = (H - H ^ (A + 1)) / A

For instance, with 3 attacks and a 20% chance of hitting (17 or more), B is ~6.6%, so a reroll is better than a +1 and worse than a +2.

With 3 attacks and a 50% chance of hitting (11 or more), B is ~14.6%, so a reroll is better than a +2 and almost as good as a +3.


The formula does not work when you get into the sure hit / sure miss range (H + B > 0.95 or < 0.05), due to auto-hit and miss rules. Still useful, though!

Thanks for correcting me. I'm glad someone here knows what they're talking about xD

[AzraelX]

I gave the thread a good read-through, and Meschlum is the guy you should be listening to. You can go through and just read his posts if you want to know the math at play here. He also explains the reason why most of the other math in the thread is wrong.

Now, like he said, rerolling will be superior for some range of values; in his example, using his assumed values, the result was that rerolling was superior versus enemies with an AC from 4 to 14.

In order to know what this range is for you specifically, more information is needed from you.

Please answer the following questions:

1) Do you have any other sources modifying your attack rolls already?

If the answer to #1 was "Yes":
1a) Is "between 8-12" the range of ACs, or the range of unmodified rolls you need to hit the ACs?

If the answer to #1a was "the range of ACs":
1b) What is the modifier already being applied to your attack rolls?

2) Do you start off knowing the AC of the enemy?

If the answer to #2 was "No":
2a) Do you find out whether your attack missed before you reroll that attack?
2b) Do you find out whether your attack missed before you roll your next attack?

[Crow]

I gave the thread a good read-through, and Meschlum is the guy you should be listening to. You can go through and just read his posts if you want to know the math at play here. He also explains the reason why most of the other math in the thread is wrong.

Now, like he said, rerolling will be superior for some range of values; in his example, using his assumed values, the result was that rerolling was superior versus enemies with an AC from 4 to 14.

In order to know what this range is for you specifically, more information is needed from you.

Please answer the following questions:

1) Do you have any other sources modifying your attack rolls already?

If the answer to #1 was "Yes":
1a) Is "between 8-12" the range of ACs, or the range of unmodified rolls you need to hit the ACs?

If the answer to #1a was "the range of ACs":
1b) What is the modifier already being applied to your attack rolls?

2) Do you start off knowing the AC of the enemy?

If the answer to #2 was "No":
2a) Do you find out whether your attack missed before you reroll that attack?
2b) Do you find out whether your attack missed before you roll your next attack?

1: Attack value is d20+24 or d20+26 (depending on circumstances) without buffs. It sits at d20+32 best case scenario, which is easy for this character to achieve.

1a: Target number to score a hit will usually be somewhere between 36 and 44 depending on target buffs.

2. In this case, yes.

[meschlum]

1: Attack value is d20+24 or d20+26 (depending on circumstances) without buffs. It sits at d20+32 best case scenario, which is easy for this character to achieve.

1a: Target number to score a hit will usually be somewhere between 36 and 44 depending on target buffs.

2. In this case, yes.

2: Then my math applies.

With a base of d20+24, you need a 12+ to hit AC 36, and 20+ to hit AC 44.

So +2 (to d20+26) is better against AC 39 to 44, and worse against AC 36 to 38.


With a base of d20+32, you need a 4+ to hit AC 36, and 12+ to hit AC 44.

So +2 (to d20+34) is worse, and remains that way until you get to opponents with AC 47 or more.


Essentially, if you'd need to roll a 15 or more to hit without the +2, taking the +2 is worth it (in terms of average number of hits) - unless you'd need a 20 even with the +2, in which case the reroll is better. If you hit on a 14 or less, then the reroll is preferable.

[Crow]

Thank you guys, for the thorough and detailed analysis. :D

[Chronos]

Looking at the average is meaningless. All that matters is your chance of hitting, and beating your needed roll by 10 is no difference than beating it by 1, but missing by 1 and hitting by 1 are completely different. So what you need to ask with any bonus or re-roll is not "by how much did this increase my roll?", but "did it increase my roll enough to turn a miss into a hit?". And yes, you do need to know the AC for that.