Save Against the Beholder Odds

IsYitzack returns, with the odds on how often the party could have saved against the beholder’s eye ray attacks!

Matt was surprised to see how well the party was rolling against his beholder on Thursday. I was wondering how surprised he should be. And the answer is very. I’ll be using DCs of 16 for all eye ray attacks (including the eyeball that came out of the column to shoot at Scanlan). The tendrils coming from the wall only required a DC 15 Dex Save to avoid. (They didn’t last long enough to require acrobatics or athletics checks to escape their grasp.) Finally, I’ll be using the Saving Throw values calculated by CritRoleStats in their character sheets.

They made 14 of 19 saving throws. That happens 15,763,665,194,354,589,813 in 2^16*10^16 events (Mathematica does arbitrary integer and fraction arithmetic so long as the RAM holds out). That is approximately 2.41% of the possible ways will come out this good. I would expect 9.82 successes (out of 19) instead. There is a 3.35% chance of doing this well or better (see Figure 1). I wouldn’t be too terribly concerned about doing this well, but it is much better than I would have expected. The most likely outcome: 10 successes at 19.3% chance of happening.

Figure 1: The plot of the likeliness of making n saves. The X axis is the number of saves (out of 19), while the Y axis is the percent chance of that many saves happening. Vox Machina in The Sunken Tomb fall very far to the right, at 14 on the X axis.

For my initial calculations, I started my computer Sunday night to begin printing out every possible combination of fails and saves of the 19 ray attacks, using the brute force method. Turns out that after reducing the probabilities from out 20 and 400 to the simplest fractions, there are still 2^16*10^16= 65,536 * 10^16 possibilities. I stopped my computer Monday morning and it was nowhere near finishing. I took inspiration from what I had done whilecalculating the odds of Vex’s resurrection ritual (Ed: coming soon!) and worked on a network of nodes to arrive at a final result in much less time.

Most of the chances of success are fairly straight-forward. Critical failures and successes would have been failures or success without being aided in any of these cases. Grog has Danger Sense which he can always use to gain advantage on Dex saves. Travis stated that “I’m not raging yet” as he rolled the dice; while he theoretically should still have it when he isn’t raging, since he didn’t claim it, I won’t claim it for him in this calculation.

Figure 2: A graph of the probability of rolling a specific number on a d20.

Figure 3: A graph of the odds of beating a DC of n. Keep in mind that DC 2 is technically the lowest possible, as 1 always fails.

Thanks again to IsYitzack for this, and the resurrection odds!