# My DM insists on rolling a single save for groups affected by AoE save spells. How does this affect my odds of successfully affecting the enemy?

As the title question, my DM rolls a single d20 save for groups affected by my area of effect spells that require a save, in order to save time. I can't help but feel like I'm being ripped off by this as a wizard with primarily AoE save-or-suck spells. I don't know if this is just a feeling or if the probabilities actually back this up. I know this can also work in my favor but it still feels off.

How are the probabilities affected when a (homogenous) group gets a single save vs. each individual in the group having their own save? I want to know specifically if this works more in my favor or more in the favor of my enemies, or if it is statistically speaking a 50/50 split. I am looking for evidence that this is a bad idea (whether it benefits me or harms me) and that the DM should roll separately for each affected target in the area of effect.

I realize this probably puts the odds in my favor when targeting weak saves in the group (i.e., WIS save on a group of ogres or orcs), but this will not always be the case and especially when there are mixed enemies in the AoE. So far we have only faced groups that contained single enemy types so I don't know what happens when there are two different enemies with two different saves.

Ideally answers will address a sliding group size (2..N group members, 5 is probably a good stopping point) and a range of save DCs -- DC 14-19 should address most levels of play.

# TL;DR

A GM shouldn't roll all-or-none saves. If reduced rolling is necessary, instead they should figure the expected number of saves, then add a d4 and subtract a d4. Below are pictured the results of this method for various numbers of enemies and probabilities of saving.

Read on to see how these are derived.

# All-or-none is a bad idea. But the GM need not roll individually, either.

## All-or-none is a different game.

We'll stipulate that in the long run (say, a thousand spells with saves) there's no difference. But we're not looking at a caster with a thousand spell slots looking at an army of goblins. We're looking at a caster with a half-dozen spell slots they'll burn before some different set of rules comes into play. (Death saves, for instance.)

The GM's all-or-none scheme makes combat more "swingy." Extreme outcomes--the caster sniping an encounter or the caster being completely ineffectual are hugely more-likely than they were before. If your GM wants to play a different game than the one in the rulebooks, that's fine. But this sort of change should be done in consultation with players, out of an agreement that you'd all like more randomness and goblin dice.

## The probability that any individual enemy saves is unchanged...

Suppose your enemy needs to roll a 9 or higher to save. There's a 60% probability of this happening after the GM says "goblin 1..." and rolls a d20. There's also a 60% probability of a successful save after the GM says "goblin 2..." and rolls a d20, and there's a 60% probability of a successful save after the GM says "I'm rolling for all goblins in the area..." and rolls a d20.

## ...but the distribution of the number of goblins saving is radically different.

Let's assume there are 5 goblins in the area and the same required roll as above. Below is pictured the probability of a number of goblins saving if each is rolled individually or if the whole group is rolled-for at once. (Click for larger image.)

Things have changed. A lot.

## All-or-not does horrible violence to mathematics.

Seriously: the GM would come closer to the "right" distribution if they just rolled a d5 to determine how many got hit. A uniform distribution would be closer to the truth than stuffing all of the probability into the two extreme cases.

Let's dig into the probabilities a bit.

First of all, the probablity that $n$ of $N$ enemies in the area make their save, when any individual enemy's probability of doing so is $p$, is given by $$P(n \text{ of } N, \text{ given } p) = \binom{n}{N} \times p^n \times (1-p)^{N-n}$$

With that in hand, let's look at how the two inputs--$N$ and $p$--really affect things.

First, let's stick with a $p$ of 0.6, and play with $N$. Here are tabulated the probabilities of some $n$ of $N$ enemies saving, for various $N$:

\begin{array}{c| c c c c c c c c} & N \\ n\text{ successes} & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\ \hline 0 & 0.00 & 0.00 & 0.00 & 0.01 & 0.03 & 0.06 & 0.16 & 0.4 \\ 1 & 0.01 & 0.02 & 0.04 & 0.08 & 0.15 & 0.29 & 0.48 & 0.6 \\ 2 & 0.04 & 0.08 & 0.14 & 0.23 & 0.35 & 0.43 & 0.36 & \\ 3 & 0.12 & 0.19 & 0.28 & 0.35 & 0.35 & 0.22 & & \\ 4 & 0.23 & 0.29 & 0.31 & 0.26 & 0.13 & & & \\ 5 & 0.28 & 0.26 & 0.19 & 0.08 & & & & \\ 6 & 0.21 & 0.13 & 0.05 & & & & & \\ 7 & 0.09 & 0.03 & & & & & & \\ 8 & 0.02 & & & & & & & \\ \end{array}

Things to notice:

• The distribution always has its peak value at the expected number of saves ($p\cdot N$, rounded to the nearest 1). The expected number of saves is also the most likely. (That's not always the case in probability, but for these scenarios it is.)
• The smaller $N$ is, the sharper that peak is. Alternatively, the larger $N$ is, the flatter the distribution is.

Second, let's look at what happens with varying $p$ values. For this we'll stick to your original $N=5$ and tabulate:

\begin{array}{c| ccccccc} & p \\ n\text{ successes} & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 \\ \hline 0 & 0.33 & 0.17 & 0.08 & 0.03 & 0.01 & 0.00 & 0.00 \\ 1 & 0.41 & 0.36 & 0.26 & 0.16 & 0.08 & 0.03 & 0.01 \\ 2 & 0.20 & 0.31 & 0.35 & 0.31 & 0.23 & 0.13 & 0.05 \\ 3 & 0.05 & 0.13 & 0.23 & 0.31 & 0.35 & 0.31 & 0.20 \\ 4 & 0.01 & 0.03 & 0.08 & 0.16 & 0.26 & 0.36 & 0.41 \\ 5 & 0.00 & 0.00 & 0.01 & 0.03 & 0.08 & 0.17 & 0.33 \\ \end{array}

Things to notice:

• Again, the distribution always has its peak value at the expected number of saves ($p\cdot N$, rounded to the nearest 1). The expected number of saves is also the most likely.
• The peak shifts up and down with $p$, and the "shoulders" around it develop asymmetries as the peak nears the ends of the distribution.

## But what to do!?

You want to convince your GM to roll all the saves. But they don't want to. Here's a better way: hand the GM a better alternative, better for them and better for you. It needs to:

• work quickly, preferably with a single roll,
• peak at the expectation value,
• be more peaked for low $N$, flatter for large $N$,
• it'd be a nice add-on to maintain the asymmetry we see as $p$ gets extreme.

### $N=1-4$

Roll the saves. The GM should have 4d20 at hand at all times--multiattack with (dis)advantage, anyone?

### $5\leq N\leq20$

The number saving is going to be simulated by

$$(\text{expected value}) + \text{d}4 - \text{d}4$$

Below are pictured some comparisons of this method to the actual distributions, were the saves rolled (click for larger images):

You can see that the outcomes generally lie within a few percent of desired. One bit of trickery: the roll at N=5 occasionally generates a negative result (because the expected value is so low). Here's how to handle those results: a result of -1 is counted as the expected value, a result of -2 is counted as the next-higher value. (That a result of -3 is nonsensical is left as an exercise to the reader. That the underlying mathematics are symmetric high-low and the same scheme works when we see results larger than N is left as an exercise to the reader.)

### Larger $N$

For N larger than 20 bump the die size to a d6. Below is pictured a result using this scheme.

For N larger than, say, 40, this sort of scheme really starts to break down--the distribution generated by dX-dX doesn't handle the flatness of fifty saves well.

I'd point out at this point--needing fifty opponents to save--you're looking at a powerful large-area spell like circle of death: you're 13th level at least. If you haven't worked this out with your GM by then, ping me.

• Well done use of clever mathematic modeling to create a pseudo-gaussian curve with a fourier row! – Trish Apr 16 '17 at 8:44
• @Trish well thanks! IME it's actually pretty easy to create quick-and-dirty simple die methods to match more-complicated results. (As long as mental math's fun for you.) The key is that if you can get the mean right, the variance in the right ballpark, and deal with a little asymmetry it's pretty hard not to be really close on the final distributions. See this answer for another fun one that I use sometimes at the table. – nitsua60 Apr 18 '17 at 14:35

Overall, this will make area/multi-target spells less reliable, but more potent.

## Numbers

The chance of a single individual to save is not affected by this change. The number of individuals affected over multiple castings is also not (or just slightly) changed.

What changes is the number of individuals in a given group that make the save. Let's assume a group of five and look at low and high chance of saving.

\begin{array}{|c|c|} \hline chance=30\% & & \\ \hline saves\ made& multi-roll & one\ roll \\ \hline 0 & 16.8\% & 70\%\\ \hline 1 & 36\% & 0\%\\ \hline 2 & 30.9\% & 0\%\\ \hline 3 & 13.2\% & 0\%\\ \hline 4 & 2.8\% & 0\%\\ \hline 5 & 0.2\% & 30\%\\ \hline chance=65\% & & \\ \hline saves\ made& multi-roll & one\ roll \\ \hline 0 & 0.5\% & 35\%\\ \hline 1 & 4.9\% & 0\%\\ \hline 2 & 18.1\% & 0\%\\ \hline 3 & 33.6\% & 0\%\\ \hline 4 & 31.2\% & 0\%\\ \hline 5 & 11.6\% & 65\%\\ \hline \end{array}

## Effect on the situation

Damage spells

If the damage is well under the individual hp of the mobs (max dmg = 60%hp or so), the change will be negligible. However, if a mob can be killed by the damage if they fail the save, we get the following (assume 50% save chance):

• multiple saves -> half of them dead, other half on half HP (most likely)
• one save -> all of them dead OR all on half HP

So the results will tend to the extremes.

Control spells

If the effect is physical (like Evard's Black Tentacles), the effect is similar to above. But with mental control spells (like Mass Suggestion) the effect will vary depending on what the GM decides. If you convince half of a group not to fight you, will this have an effect on the others? I would say that the one-roll method will generally make mental control spells more powerful, as half of the effect is often not enough.

## An alternative

If your GM does not want to roll every save, he can assume the number of saves made is the expected value. So if they save with probability p, and there are n enemies, p*n of them make the save and the others do not.

An example with concrete numbers: a member of the group makes the save 40% if the time (+1 vs DC14 lets say), and there are 5 of them, then 5*40% = 2 makes the save, the remaining 3 does not.

The same calculation from a different perspective is used for handling mobs in the DMG (250). Instead of AC and succesful attacks, just use DC and successful saves. You can use that if you like looking things up in tables more than calculating from scratch :) (Thanks to daze413 for bringing it up.)

• One important argument against is that it will trivialise some encounters (those where the monsters fail to save). Instead of making the fight a bit easier (e.g. disabling half of the monsters), they all get the or-suck treatment and the fight is over. It makes many fights binary, decided by a single save roll. – Mala Mar 20 '17 at 19:56
• I don't know if I'm just missing something due to not being very familiar with D&D, but: isn't n surviving enemies after any given round inherently mean more damage/harm/problems for you or your party than < n surviving enemies? This is why you focus-fire in RTS games for example - sure, if you just consider your side, you're doing the same damage rate overall, but shouldn't we also keep in mind that every enemy that survives longer is that much more damage pointed at you for each additional round? In that sense, single-roll-for-AoE approach seems to me more harmful than indicated here. – mtraceur Mar 23 '17 at 12:00
• @mtraceur I agree with you. Take a look at Yakk's answer, he deals with this aspect. I decided not to lift his idea into my own, just upvoted his :) – Szega Mar 23 '17 at 17:45

# This is NOT in your favor

What you are experiencing is called "risk aversion" -- you intuitively sense the losses of this trade more than the gains. And you are right to feel this way.

However, before we get into if this is good or bad for you, I'll make some assertions. The save DC doesn't matter, and the target's save modifier doesn't matter; at least, not individually. What matters is the target on the die.

So, for example, with a DC 14 Wisdom save against a +2 Wis save modifier, they have to roll a 12 to save. This is the same as a DC 15 Wisdom save versus a +3 save mod, or a DC 11 Wisdom save versus a -1 save mod. That is, it's the probability that matters.

The reason that we want to look at it this way is because it reduces the saving throw into a weighted coin toss. From an entire array of save DC's and save mods, the end result is equivalent to what happens if we flip a loaded coin. In the above example, if you were betting on heads they lose, we're flipping a 45 (tails)/55 (heads) coin.

This approach has an advantage. Because we're thinking of saving throws as coin tosses now, we can apply analogies to test our perspective that wouldn't be easy to apply to a die roll.

# Analogies of Dice and Coins

This 7 minute video features a man going on the streets and offering people $10 of his money against$10 of random strangers' money on a coin toss, and seeing how they react. From the video, he seems to always be rejected. The reason seems to be because the participants don't want to lose their $10. They simply didn't see the gains of an extra$10 as enough to offset their risk aversion.

The man then ups the ante. He offers $12 to some, to others$15, and to others $20. Many still reject him, though this statistically puts the bet in their favor. The perceived losses still outweigh the perceived gains. And then he offers a solution: repeat the bet of his$12 to their $10 on a coin toss 10 times. Now, some of the participants begin changing their minds, though they aren't able to explain why. I recommend watching the video beginning at 3:03 to 4:45, where he explains why repeating the coin toss when it is weighted favorably to one side is much more favorable than when the toss is done only once. Here's the gist: when you do only one toss, the bet could go either way. Even when it's in your favor, it's still only a single coin toss and you could still lose. But when the toss is repeated multiple times and the bet is weighted in your favor, you are almost guaranteed to win. In the example in the video, putting up a$20-to-$10 bet on a fair coin toss, there is only a 1-in-2300 chance of losing money in that scenario for the side whom the bet favors, whereas there is still a 1-in-2 chance of losing money for the same bet if it was done only once. So: when the coin is flipped multiple times, it is in your favor if the bet favors you. # Let's go back to D&D Now, whenever you cast a spell in D&D, the bet favors you. What you are putting up is your spell slot (your hypothetical$10 you stand to lose), and what they are putting up is the targets experiencing the spell's effects (their hypothetical $20 you stand to gain). And when the DM rolls the saving throw, it is akin to flipping a weighted coin that favors you (as if flipping a coin with a >50% chance to go your way). So: when you cast a spell, you're making a bet that favors you. And then the DM "flips a coin". Let us define what it means to win or lose this bet. • When you lose this bet, you completely wasted your spell slot and got nothing in return for it. Alternatively, your targets all took half damage because they all passed -- surely all targets passing the save is a failure to you as a caster. • When you win this bet, you do not waste your spell slot because some of the targets will fail their save. • So: you are going into this bet knowing that some of the targets will pass, and that's fine with you as long as most of the targets fail. If the DM flips this coin only once, then you stand to lose the bet with somewhere between a 1-in-2 through 1-in-5 chance depending on the exact probabilities. However, if the DM flips this coin multiple times, once for each target, then you are almost guaranteed to win. If there were three targets, for instance, and they had a +3 save mod versus your DC 18, the odds of you losing are 1-in-37, as opposed to an odds of 1-in-3 to lose the bet if the DM only rolled once. # Why this is bad for you The DM has now increased the odds of you wasting your resources. The RAW gives you a virtual guarantee that you will not lose your spell slot when you use it to cast an AoE spell. Your DM has overruled that and made it so you often can lose it. In the example of a DC 18 versus their +3 save mod, and the AoE targeting three creatures: your DM has made you 12 times more likely to lose as the odds went from 1-in-37 to 1-in-3. And of course, this will make anyone feel cheated. # Failure Case: Reframed Instead of thinking in terms of "success" and "failure", ask yourself instead: how has the DM made a miniscule chance more probable? That is, when rolling multiple die, there is only a small chance all targets pass the save. How has moving to a single die roll affected the odds, and made it more likely for all of them to actually pass? # Quantifying Changes in the odds Consider this scenario: your save DC is 19, the five targets have a -1 save modifier. You cast the spell, and so they need to roll a 20 to make it. The DM rolls a natural 20. They all pass. Does that make you feel like the spell slot is wasted? If so, then the above table is useful for you. The above table shows the change in odds depending on the target on die and the number of creatures affected. Previously, the odds were miniscule that all targets passed that save, but this change has amplified that chance. In fact, the odds are 160,000 times more likely now that all 5 targets pass their save if they need to roll a 20. In my previous example of a DC 18 save versus a +3 save mod against 3 targets, the target on die is 15 and there are 3 targets. That is the 6th row, 2nd column, which shows that the odds are 11.11 times more likely for all targets to pass their save now under the single die roll regime (I rounded off to 12 above). You might note the other side of the picture: but now they are also much more likely to all fail their save. If you are willing to take that risk, then you will not feel cheated when the DM changes the multiple die rolls to a single die roll. It is up to you to decide how much risk you are comfortable with, and how OK you are with accepting that the result of your spells will be very swingy/extreme. # Final Words Regardless of the method of rolling for saves, whether a single die or multiple die, casting an AoE spell is still in your favor. A premise of this answer is that spellcasting is akin to betting your$10 against their \$20 and flipping a loaded coin that favors you. So no matter what, in the long run, your AoE spells will still do more good than bad.

However, the difference between tossing a coin once, and tossing a coin 10 times, is not negligible. It affects your odds of winning and losing every time you flip the coin. If you are comfortable with the higher swing, then there is no cause for concern.

However, as far as this question is concerned, we want to provide evidence to overturn the ruling and move it back to rolling multiple die for saves. This implies a low appetite for risk. And a low appetite for risk is concerned about the odds of loss.

I hope you are able to use this answer and convince your DM to see your way.

# Response to Objections

1. This does not show the entire argument as it only calculates loss. What about the gains? What about the increased chance they all fail the save together?

I am not arguing from a long term perspective. Over the long run, the method their DM chooses does not matter as the same number of creatures will fail their saves whether we went with multiple rolls or a single roll to determine the creatures' saves. In other words, the long term perspective is immaterial. What I'm showing is, how does this change things for the player now? And the answer is, it changes the odds now, and they stand to lose more often as a result.

2. Isn't this answer mathematically incorrect?

Feel free to verify the math and show how it is wrong, so I can update this answer. As for my part:

Given a save chance of, for example, 30% (aforementioned DC 18 vs +3 save mod), to get the odds of failure for multiple die rolls: 1-to-1/(0.3^n), where n is the number of creatures. In this example, it was n=3, so the odds were 1-to-37.

The odds of failure on the single die roll case: 1-to-1/(0.3), or 1-to-3

Compare the two odds. 1-to-37 is 12 times less likely to fail as 1-to-3

3. Why doesn't this answer acknowledge that there are benefits to be gained from this ruling?

Pulling arguments from the other comments who have contributed to this discussion, a swingy game means more character death. The DM loses nothing when they roll once, but the player loses a spell slot when the targets save. The DM has many NPCs who can suffer the negative consequences of all targets failing their save, but the player's character can die. In other systems, swingy dice results means the players go into such games expecting their characters to die. D&D is not such a game (at least, not 5e).

Discussing risk aversion and why we avoid risk is a thing people do in real life. It's a thing that can be done in D&D too, apparently.

The benefits to be gained by the player (ie: that they can have all creatures fail their save at the same time) is not valuable in the context of the question raised, otherwise this question would not get asked in the first place. We feel loss more strongly than we feel gains, and ten good moments in the session may feel clouded by that one bad thing that happened. The fact is: if you burn a spell slot and all creatures passed that save, that sucks for you as an AoE caster.

4. This definition of success/failure is wrong

There are many ways we can define failure. You can say that it is not necessarily a success if only 1 creature fails the save. But that is void of context. How many creatures were there, two? Three? Four? At which point do we draw the line between success and failure for when the case is not binary? This is not a math question, but a player one, and the answer depends on you and your risk appetite. The math will change depending on your definition, but your definition is personal to you, and this answer is predicated on a low appetite for risk (as is made apparent by the way I formulated my definitions).

We can discuss what should be the right way to categorize success/failure when talking about rolling multiple saves on many creatures, but what's the point? How can the player approach his DM with this information and overturn the ruling?

If you are personally good with casting a spell and having all targets pass the save, then that is good on you. But this answer does not include your perspective, because if we adopt your perspective, then there is no problem. It is concerned about the people who aren't alright with that, because that is where the problem exists.

And that is why I use the failure definition of failure occurring when all targets pass the save: because it is not a contestable case (ie, you can't call that a win). And the players who care about getting a sort of guarantee that they will get some utility out of their AoE spells, those will be the people who have a low appetite for risk, too.

• It should be brought up that rolling once gives you a much larger up-side. All creatures take damage if their ST fails. – Shem Mar 20 '17 at 18:10
• -1 The reasoning is spurious. It answers "why do you erroneously feel like this is a bad deal" instead of whether, mathematically, it is. – mxyzplk - SE stop being evil Mar 20 '17 at 19:07
• @mxyzplk I've just had this entire argument with Szega in the comments, and have those been deleted. All I can say is: your gut feeling is wrong. – user27327 Mar 20 '17 at 23:41
• The video posted as part of this response actually argues against the poster's argument that rolling once per group is somehow worse than rolling for each individual. The error he makes is that he applies the repetition @ group vs individual, instead, as the video shows, one group vs. many groups. Over a large number of groups, the percentage that save as a group will resemble the percentage that would save individually, and this resemblance will approach exactness the more the sample size grows. – Be Cn Mar 21 '17 at 21:16
• @BeCn "Over a large number of groups, the percentage that save as a group will resemble the percentage" is entirely true but is beside the point. That long term exactness does not make a single use of a single spell in a single trial against a group any more attractive; in fact, it makes it less attractive due to the lopsided risk vs reward. – LegendaryDude Mar 22 '17 at 21:01

This would be a great change if you were in a fight where you were almost certainly going to lose.

Imagine a spell that defeats 1/2 of the creatures it targets. You use it on 2000 creatures. If more than 10 survive after the spell is cast, you lose.

With one roll, you have a 50% chance of winning the fight.

With one roll per creature, you certainly lose the fight.

In most D&D fights, the players are on the side that is going to win more often. When you are on the side that is more likely to win than the other, usually anything that increases reliability (decreases variance) boosts your chance to win, and anything that increases variance (decreases reliability) is going to lower your chance to win.

One roll for all foes increases variance. Instead of a bell curve centered around (chance to fail * number of foes), you get a bimodal distribution with (everyone fails) at chance to fail, and (everyone succeeds) at chance to succeed.

While the average remains the same, you don't actually care much about the average unless you are repeatedly doing something and you care about the sum.

There is another way to look at it. Numbers of foes are usually super-linear in effect. Twice as many foes is more than twice as hard to fight, barring AOE effects, because while you are defeating the first N foes the second N foes are all beating on you.

A spell that defeats half the foes (approx) thus makes them almost 4 times weaker. Half as many foes die twice as quick, and deal about half as much damage each round. Half times half is about 1/4. (The exact values are a bit different -- a triangle rather than a square -- but this is rougly correct).

A spell that half the time defeats all the foes, and half the time defeats none of the foes, makes them on average half as strong.

Half is bigger than a quarter. So you'd rather defeat half of them, than have a 50% chance to defeat them all.

Now, there are some situations where instant-win is a reward; where the fight is going to go poorly even if half of them remain, basically.

On top of the "you could not defeat half of them" underdog situation, imagine a situation where there are guards able to set up an alarm. A "you all lose with 50% chance" spell becomes a 50% chance to avoid the alarm. Taking out half means "the alarm is going off".

However, in traditional D&D-esque situations, where parties are expected to succeed at what they are doing, and failure is something that comes along relatively rarely, this is a significant reduction in the power of said spells.

I get the desire to speed this up. But, rolling a fistful of d20s with the same target number and counting is pretty fast. You can even let the DM pick which ones fail, and you'd usually be better off than the "one roll to rule them all" rule.

This is not so much the question of mathematics, as what makes a game interesting.

## Extreme examples

Banishment
3 Fire Giants charge your level 11 party, your spell DC is 17, their Cha save is 5. If you cast Banishment at 6th level, targeting all of them, the best case scenario is that 1 or 2 of them is removed from the encounter while you finish off the rest. When you are done and arrange the party in position, you stop concentrating, and retrieve the remaining Giants.
Bad is sending them all away, so the party has nothing to do except moving into position, and worst is all of them stays, you just wasted a slot and action.

With a usual DM, you have more than 74% chance for the best outcome, 16.6 for the bad and only 9.1 for the worst.
With your DM, you have 0%, 55% and 45%, respectively.

Hypnotic Pattern
More or less the same, just the percentages differ, as their Wis save is different. If you "turn off" half of them, you get an easier encounter, and that is what would happen with most DMs. With your DM, you either affect all of them creating a boring encounter, or none of them, now the encounter is quite a lot harder, considering the Wizard just wasted a turn and a slot.
Neither outcome is really desirable.

# Conclusion

Statistics says in the long run the numbers even out, during 1000 encounters you affect the same number of creatures with or without this houserule. However, each individual encounter will be hurt by this in my opinion.
Encounters either become too easy, or too hard. Neither is good for you or your party.

You should ask your DM to stop, I think the few seconds he spares are not worth the above outlined problems.

• This is the only argument here that looked at the important side first. Regardless of whether the math works or not, it sounds like the player isn't having a lot of fun, there's a problem, and the GM needs to work something out for that player. Or, if not, the player should probably get in a game where he can have fun. – Be Cn Mar 21 '17 at 21:23

This is simply answered by considering the following truism:

More randomness in encounters hurts the PCs more than it hurts the monsters!

This is because the PCs are in every encounter the monsters are usually only in one: the one where the PCs kill them. As such, the PCs will see all possible outcomes of that randomness, including the most extreme results (good and bad), the monsters will generally only see the more typical results. Even though they see the "good" extremes as often as the "bad" extremes this is to their overall detriment because bad extremes can kill them!

For a group of $n$ creatures with a probability of saving of $p$.

From the binomial distribution, for individual rolls the mean number of successes is $np$ with a variance of $np(1-p)$.

The single roll is equivalent to a binomial distribution with $n=1$ multiplied by the number of creatures. This has the same mean $np$ but, from the basic properties of variance, a variance of $n^2p(1-p)$. Since $n \ge 1$ always, $n^2\ge n$ always. That is, your DM's way of doing things has a greater variance which means, in non-technical terms, is "more random".

It follows that this hurts you more than it does the monsters.

I agree that this rule is not in your favor.

Szega's answer covers the probability distribution pretty well, and I believe his math is sound. Markovchain's math seems sound, and he generally agrees with Szega, but I think his conclusions are slightly off. I still generally agree with him. To summarize the points they agree on, when your GM rolls a group save, the expected value of the damage you do does not change, but an extreme outcome is guaranteed.

I believe that this rule is not in your favor because extreme bad outcomes are much worse than the median outcome, but extreme good outcomes are only slightly better than median. When you get very unlucky, you die. When you get very lucky, you use fewer resources.

Take an example where you throw a fireball at a group of orcs. With your GM's rules, the spell does either half or full damage. Your party either still has a tough fight, or they mop up the remainder. For a single fireball this is not a huge deal.

Lets extend this example. You are in the final fight against an especially tough group of orcs. You throw a series of fireballs. With the standard rules, there is a very high probability that the damage your fireballs do is somewhat close to the expected value. With your GM's rules, the orcs will probably fail some and succeed some, but the chances of succeeding or failing ALL their saves is much higher than standard. In a boss fight, you can get TPKed if they save all their rolls. If they fail all their saves, the fight ends slighly sooner.

You have more to lose from an extreme bad outcome than you have to gain from an extreme good one.

The short story is that for this probability distribution, the mean (first central moment) stays the same, while the variance (second central moment) is increased.

So the results are more "swingy". You'll have some encounters where every enemy falls to your spell, and all the rest in which no enemies fall to your spell. The latter will of course be more dangerous. As others have pointed out, as a general rule of thumb increased encounter variance works against the PCs in the long run, as more encounters provide an opportunity for a TPK.

In some sense, the enemies who pass their saves against your spells will all be "batched up" together in units opposed against you, to their advantage. As a player, if I had an option, I would choose the lower-risk path where every enemy makes an independent saving throw; that is: allowing you to divide-and-conquer a series of opponents. You might suggest to your DM that some other DMs roll a fistful of d20's for saves in situations like these.

• I think the specter of TPK isn't necessary: even one character death would be a bad enough outcome to prefer playing by the rules to this GM's houserule. – nitsua60 Mar 21 '17 at 15:43

The answer depends on what spells you are casting and why. You can tailor your spell choices to turn this in your favor.

For most spells/situations, the benefit gained is proportional to the number of targets effected. In which case, other answers are correct that you generally want to minimise the randomness and have each monster save separately.

For some spells/scenarios, there are diminishing returns with number of targets impacted. Stunning half a group is likely to be considerably more than 50% the benefit of stunning the whole group as it locks the group out from using synergies. Slowing half a group can sometimes be better than slowing the whole group as you can separate them.

For some spells/scenarios, there are increasing returns as the number of targets unaffected increases. This is usually the case where an unaffected target may be able to mitigate the harm done to other targets. An illusion that convinces 100% of targets is likely to provide more than twice the value of an illusion that is seen through by half the targets. Anything that effects memory or thought is likely in the same boat. Dispel 'fly' on 5/10 targets, and you have 5 creatures diving to catch their 5 friends. Dispel fly on 10/10 targets and you have 10 corpses to loot.

Even with pure damage, there are times when hitting 100% of targets is much better than hitting 99% of targets. If one target survives, they can raise the alarm, escape with vital information, kill the hostage etc.

• Good counter-examples to show that increased variance can benefit the player in some circumstances, including pure damage spells. I was completely convinced the all-or-nothing rule was completely unplayable before reading your analysis. Now I think it fundamentally alters how you build your character, prepare for encounters. and play the game, but that it's an interesting house-rule (with consent from the whole group) instead of a sucky shortcut. – stevenjackson121 Mar 21 '17 at 16:49