This is a follow-up article to “Progressive Dice for Effects Durations,” in which I propose a method to roll each turn for the chance for an effect to end. This, in order to maintain the secrecy—and suspense—of an effect’s duration when playing solo or otherwise without a DM.

So-Called “Progressive” Dice

“Instead of rolling for the effect duration at the trigger to know on what turn the effect ends, we roll the same dice at the beginning of each subsequent turn to see if the effect ends in that turn.

“A dice result equal to or greater than the current subsequent turn indicates the effect continues. Roll again at the beginning of the next turn. A lower result means the effect ends.”

—“Progressive Dice for Effects Durations”

I’ve used progressive dice for effects durations and a number of other things for years. My assumption was that the chance for the effect to end each turn stands alone turn by turn, increasing as turns go by, therefore “progressive.” I also assumed that the overall probabilities, compared to the traditional method, were somehow the same.

Writing the previous article forced me to think a little deeper on the method. I wondered if I’d got it right. Does it really yield a progressive chance, turn by turn, for the effect to end?

The smartest D&D mathematician I know is Dan Collins of Wandering DMs and Delta’s D&D Hotspot. A 40+ year D&D veteran, Dan is also a university lecturer in mathematics and computer science.

So, I sent him a query outlining the problem. Dan’s response, a few lines and a table of probabilities, shows how it is that progressive dice are not so progressive after all. For, using the proposed method, the chance of the effect ending is much higher in the initial periods than the later, so, not at all statistically equivalent to the traditional method.

In a traditional game, the DM rolls a single dice (or combination thereof) when an effect is triggered to determine its duration. A duration of 1 to 6 turns, say, is rolled on a d6. The probability that the effect ends on any turn is ⅙ or 16.67%.

Using so-called progressive dice, “It stacks up differently,” Dan writes. “It’s very unlikely that you’ll get to turn 5 or 6, because you have to survive all the prior rolls to get there. Over half the time you’ll have the effect stop after two or three rounds.”

Here I had to make a saving throw vs. Death Ray. Reading the email, I was talking to Dan through the screen: The progressive dice method is so elegant, man—it has to be right!

Dan goes on to explain: “Computing a compound probability like this is a series of multiplications…” He also includes a table with a note that, if the calculations are correct, the sum of all chances should be 100%. I reproduce the table here.

So, it isn’t just the simple chance (bold) each turn that the effect will end. We have also to factor in the cumulative chance (italics), which is each previous roll inverted, that the effect hasn’t already ended.

Note that, in the previous article, we roll to see if the effect ends at the beginning of the next turn. “Ends after turn n” is a different way to say the same thing.

Therefore, at best, I misnamed “progressive dice.” Though the number to roll increases turn by turn, the chance to make that number is not at all progressive. The chance to end the effect after the second or third turn is much higher than the first or later turns.

Alternatives

So, what is a DM-less player to do? We might accept the statistical difference and use the so-called “progressive” dice in play. Or we might seek out other solutions. We look here at two—one of them works.

Single Dice, Effect Ends on a 1 (Not a Solution)

I thought of an alternative method. Roll the same dice every turn, with a result of 1 signaling the effect’s end. The effect ends automatically at the end of the  maximum duration.

It’s more simple than counting turns. But, if I’m following Dan’s lesson well, we still have to factor in the chance that the effect ended with the previous roll(s).

Furthermore, I note that the total percentage is only 66.51, which is 33.49 short of 100. I’m guessing that’s because the effect automatically ends at the duration’s upper limit. The chance that it will end after the 6th turn is, in fact, 6.70 plus the remaining 33.49, or 40.19%.

1 to n Cards

Dan suggests a card solution: a number of playing cards n equal to the upper limit of the range, 1 to n, one of which is an ace—or, if you have a deck of many things on hand, the Donjon (ace of spades) or the Fates (ace of hearts).

Shuffle the deck once when the effect is triggered. Draw one card from the top of the deck at the beginning of each turn. When the ace comes up, the effect ends.

Here, the shuffling is the dice roll, which determines on which turn the effect ends (on the ace). The chance that it will end in any particular round is ¹⁄_n, just like a single dice roll. The only practical difference from the dice roll is that the ending turn, while predetermined, is hidden within the deck. Also elegant.

A disadvantage is that the card method cannot duplicate dice combinations. Melqart’s stun duration, 2d4 turns, for example, cannot be reproduced using this method. In this case, it was the first effect duration of the campaign, but dice combinations might be infrequent.

Another disadvantage is that you have to manage an additional tool at the table. The suspense about when the effect ends, though, may well be worth the trouble.

For myself, I love to incorporate playing cards into my D&D, and if there’s an opportunity to get more use out of a deck of many things, I’ll take it.

Other Solutions?

I’m interested to hear your suggestions for maintaining the secrecy of effects durations in a DM-less game. I would also entertain a counterargument showing that progressive dice do in fact produce progressive results. Because it’s elegant, man, it has to be right!

My thanks to Dan Collins for his statistical analysis of the problem as well as an alternative solution. For interested readers, Dan offers several venues to learn more about dice and probability. In an episode of Wandering DMs, Dan gives a course in Basic Dice Math, and in another episode with cohort Paul Siegel, he talks Dice Mechanics. In addition, you’ll find a plethora of articles about dice statistics on Dan’s blog.

In the first foray into the Deep Halls, Melqart is stunned by the defensive explosions of scarab beetles. The effect lasts for 2 to 8 turns.

Normally, the DM rolls 2d4 and makes note of the turn on which the character recovers. Playing solo or otherwise without a DM, though, we should not know when the effect is to wear off.

In Melqart’s case, had I rolled the variable duration immediately, I might be tempted to plan the next turns—or otherwise use the information unconsciously. “We guard Melqart until he can move again…” This breaks the narrative tension and challenges verisimilitude.

Procedure

For these occasions, I use what I call progressive dice. Instead of rolling for the effect duration at the trigger to know on what turn the effect ends, we roll the same dice at the beginning of each subsequent turn to see if the effect ends in that turn.

A dice result equal to or greater than the current subsequent turn indicates the effect continues. Roll again at the beginning of the next turn. A lower result means the effect ends.

Examples

Simple Variable Duration

A shrieker’s alarm sounds for 1 to 3 rounds after exposure to light. The next round is the first round of the effect duration. No need to roll this round, as the effect continues even on a 1. On the second subsequent round, the shrieking continues on a 2 or higher. Third round, the shrieking continues only on 3 result. In which case, it ends at the beginning of the next round, having reached its maximum duration.

Fixed Plus Variable Duration

A character quaffs an invisibility potion, which lasts a fixed period of 6 turns plus a variable duration of 1 to 6 turns (by my reading of Holmes, 37), which is 7 to 12 turns. For the first 7 turns, no roll is necessary. The character is invisible. At the beginning of the eighth turn—that is, the second turn of the variable duration—roll a d6. On a 2 or higher, the invisibility effect continues. Less than 2, the effect ends; the character becomes visible.

Any Dice Combination

Melqart is stunned for 2 to 8 turns. Roll the same dice combination, 2d4, at the beginning of each subsequent turn, ignoring the first and second, when the result can only indicate the effect continues. But count all turns following the trigger as subsequent turns. At the beginning of the third turn of the duration, a 2d4 result of 3 or greater means the effect continues.

In Play

Disadvantage

More Dice Rolls: We are effectively replacing a single dice roll with a series of rolls, which we like to avoid as it takes more time.

Advantages

Fewer Notes: On the other hand, rolling the usual way, the DM must note and remember when the effect will end. Rolling progressive dice, at least one player has a vested interest in the roll, so it isn’t easily forgotten.

Player Agency: Even with a DM, the player may be allowed to roll the progressive dice.

Increasing Tension: There can be a lot riding on that dice roll. As the turns pass, the tension mounts.

While Melqart squirms on the floor, moaning, palms over ringing ears, the harpy is leading the rest of the party—all charmed—to its nest. Will Melqart come to his senses in time to save them…?

Statistically Equivalent?

I am uncertain whether there is a difference, using progressive dice, in the statistical chance for the effect to end in any particular point in the duration.

Rolling in the usual way, we have a 1 in 6 chance for the effect to end after any of six turns. At the same time, there is a 100% chance (6 in 6) for the effect to last at least one turn, a 5 in 6 chance it will last at least two turns, 4 in 6 for three turns, and so on.

With progressive dice, the chance to end the effect increases as the turns go by, starting at 0 in 6 in the first subsequent turn, 1 in 6 in the second turn, and so on, up to 5 in 6 at the beginning of the turn of maximum duration. If it doesn’t end on this turn, it will certainly end at the beginning of the next turn.

Intuition tells me it’s the same chance, but that guy has been wrong before. I’ve put the question to a smart math person. Your comments are also welcome. I’ll add an update when I get something.

Once again intuition leads me astray. The so-called “progressive” dice method described in this article is not statistically equivalent to the traditional method. This method is still useful in play, provided we accept the limitation. For further explanation and an alternative solution, see “Progressive Dice, a Misnomer.” [08:30 21 January 2022 GMT]