Is there an easy formula for multiple saving throws?

Question

My issue: I am playing a druid and if I do something like summon 8 wolves I can end up forcing the DM into 8 saving throws each turn.

I mitigate this on my end with an app to roll lots of dice, I pre-roll my attacks and damage and all that, so my turns are quick, but the DM mentioned after the game that he doesn't like having to roll so many saving throws, and worries it might slow things down.

Is there a reasonably simple formula to convert multiple saving throws into a single dice roll?

Example: 8 DC12 Strength saving throws becomes a single DC18 save - the outcome being that the target has a similar % chance of overall success, but with less dice.

My expectation is that given there are various throws of various difficulty, and each creature will have a different bonus that what I am asking for simply can't be done, but I don't have the maths to prove that.

What I am not looking for are ways to convince the DM that it isn't too bad (It is no different than a well placed AoE for example), or to not summon so many wolves, spread the attacks around, or that I don't get to pick what I summon, or any of the other usual advice about saving table time. My turns take seconds because I plan in advance, and I don't think the DM wants to change what they do (IE: no using a dice roller for them), so this is just me looking to answer this one question.

Answer 1

Yes, sort of, but maybe not really

Basically, the odds for a creature passing one save (no advantage or disadvantage) is

$$P(\text{DC} < \text{d20}+\text{Bonus})$$

If we move the bonus to the left to reduce the DC instead (same difference) we see that this is simply the odds of the d20 getting higher than a certain number (if DC≤1 the probability is 1, if it’s greater than 20 it’s 0). So assuming $$1 \lt \text{DC}-\text{Bonus}\leq20$$ this is simply $$1-\frac{\text{DC}-\text{Bonus}-1}{20}$$ (the -1 is because on a DC 1 a 1 is a success). This’ll give you a number which you can just look up on this table:

$$ \begin{array}{l|l} \text{DC}-\text{Bonus}&\text{Success Probability}\\\hline 1 & 1.00\\ 2 & 0.95\\ 3 & 0.90\\ 4 & 0.85\\ 5 & 0.80\\ 6 & 0.75\\ 7 & 0.70\\ 8 & 0.65\\ 9 & 0.60\\ 10 & 0.55\\ 11 & 0.50\\ 12 & 0.45\\ 13 & 0.40\\ 14 & 0.35\\ 15 & 0.30\\ 16 & 0.25\\ 17 & 0.20\\ 18 & 0.15\\ 19 & 0.10\\ 20 & 0.05\\ \end{array} $$

Now we know the chances of one of these attempts succeeding. Now, to get the odds for all of them succeeding we simply have to raise this chance to the n-th power, where n is the number of attempts (i.e. attacking wolves in your example). So for example, if we have a wolf that imposes a DC 12 strength save and the creature has +3 to strength save, we look up 12-3=9 in the table and get a probability of 0.6 that they succeed in their save attempt. However, the chances of them succeeding 8 times are now

$$0.6^8 \approx 0.017$$

Here we see the problem with this approach though: We get the odds, but how do we actually simulate these using dice? Basically, the best thing we could do here is use per-myriad dice (i.e. use percentile dice but 4 of them instead of 2 to get 4 digits). This is less dice rolling than rolling all 8 saves (technically you could also get a 10,000 sided die, but that’d be practically indistinguishable from a sphere at that point), but I’m not sure it’s enough of an improvement to be worth it. Anyway, here’s a table with the success probabilities for up to 8 creatures. As you can see the odds for passing so many DCs in a row get really tiny especially as DCs increase, so at some point you’d have to use even more percentile dice (up to 6 if you want to use this table) to do this.

$$ \begin{array}{l|c|c|c|c|c|c|c|c} \text{DC-Bonus} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\\hline 1 & 1.000000 & 1.000000 & 1.000000 & 1.000000 & 1.000000 & 1.000000 & 1.000000 & 1.000000\\ 2 & 0.950000 & 0.902500 & 0.857375 & 0.814506 & 0.773781 & 0.735092 & 0.698337 & 0.663420\\ 3 & 0.900000 & 0.810000 & 0.729000 & 0.656100 & 0.590490 & 0.531441 & 0.478297 & 0.430467\\ 4 & 0.850000 & 0.722500 & 0.614125 & 0.522006 & 0.443705 & 0.377150 & 0.320577 & 0.272491\\ 5 & 0.800000 & 0.640000 & 0.512000 & 0.409600 & 0.327680 & 0.262144 & 0.209715 & 0.167772\\ 6 & 0.750000 & 0.562500 & 0.421875 & 0.316406 & 0.237305 & 0.177979 & 0.133484 & 0.100113\\ 7 & 0.700000 & 0.490000 & 0.343000 & 0.240100 & 0.168070 & 0.117649 & 0.082354 & 0.057648\\ 8 & 0.650000 & 0.422500 & 0.274625 & 0.178506 & 0.116029 & 0.075419 & 0.049022 & 0.031864\\ 9 & 0.600000 & 0.360000 & 0.216000 & 0.129600 & 0.077760 & 0.046656 & 0.027994 & 0.016796\\ 10 & 0.550000 & 0.302500 & 0.166375 & 0.091506 & 0.050328 & 0.027681 & 0.015224 & 0.008373\\ 11 & 0.500000 & 0.250000 & 0.125000 & 0.062500 & 0.031250 & 0.015625 & 0.007812 & 0.003906\\ 12 & 0.450000 & 0.202500 & 0.091125 & 0.041006 & 0.018453 & 0.008304 & 0.003737 & 0.001682\\ 13 & 0.400000 & 0.160000 & 0.064000 & 0.025600 & 0.010240 & 0.004096 & 0.001638 & 0.000655\\ 14 & 0.350000 & 0.122500 & 0.042875 & 0.015006 & 0.005252 & 0.001838 & 0.000643 & 0.000225\\ 15 & 0.300000 & 0.090000 & 0.027000 & 0.008100 & 0.002430 & 0.000729 & 0.000219 & 0.000066\\ 16 & 0.250000 & 0.062500 & 0.015625 & 0.003906 & 0.000977 & 0.000244 & 0.000061 & 0.000015\\ 17 & 0.200000 & 0.040000 & 0.008000 & 0.001600 & 0.000320 & 0.000064 & 0.000013 & 0.000003\\ 18 & 0.150000 & 0.022500 & 0.003375 & 0.000506 & 0.000076 & 0.000011 & 0.000002 & < 0.000001\\ 19 & 0.100000 & 0.010000 & 0.001000 & 0.000100 & 0.000010 & 0.000001 & <0.000001 & <0.000001\\ 20 & 0.050000 & 0.002500 & 0.000125 & 0.000006 & <0.000001 & <0.000001 & <0.000001 & <0.000001\\ \end{array} $$

All that said there’s a much simpler way

Your DM is already OK with you prerolling attack rolls. There’s no reason you can’t preroll saving throws too. Yes, you’d be rolling for the attacked creature, but unless this creature has special abilities interacting with saving throws it shouldn’t really matter if you or your DM roll the dice, so you can just tell him which rolls you got and they can just check if any of them fail the DC the same way he checks your attack rolls against the creatures AC.