Kinda-Exploding Dice Mechanic

Saelorn

Adventurer
I saw a game recently which used your typical mechanic where you roll a bunch of dice and count successes. This game used d8s, instead of d6s or d10s.

The interesting thing was, instead of saying that a roll of 6 or 7 counted as a success, and a roll of 8 counted as two successes; it said that a 6 or 7 was a success, and an 8 was also a success, but if you rolled an 8 then you also rolled two more dice, which could also be failures or successes, or could also be an 8 that was a success where you rolled two more dice, and so on.

The part that wasn't immediately intuitive to me, was what the average success rate is supposed to be. If I'm rolling ten dice, how many successes should I expect?

With something like Exalted, you roll d10s and 7 through 10 is a success, but a 10 is two successes - you roll ten dice, you'll probably get five successes. This system is more complicated than that, and I'm just not seeing where the math ends up. Is it the same? Do you average out to rolling a number of successes equal to half the number of dice being rolled?
 

GoblinKid

Visitor
Each D8 has a two in eight, or one in four chance of rolling a success, and a one in eight chance of rolling another two die with a success, whereas each D10 has a three in ten chance of success, with an additional chance of rolling two successes. this can be simplified as having a two in five chance of getting at least one success. So:
D10 : 40% chance of success
D8 : 37.5% chance of success

The benefit to the D8 in this case though, is that you can always succeed at a task, if your lucky enough.
 

Saelorn

Adventurer
Thanks for the response. I get that there's a 3-in-8 chance of getting at least one success on each d8, but there's also a chance of getting two or more successes on each d8. I'm trying to find out the average expected number of success to get from each die.

There's a 37.5% chance that the first roll of a d8 will be a success.

The chance that rolling one die would result in more than one success is equal to the chance of rolling an 8 on the first die (12.5%) times the chance of rolling at least one success on one of the two explosion dice.

Of the 64 possible outcomes of rolling 2d8, there are:

25 combinations which result in no additional successes.
20 combinations which result in one more success.
10 combinations which result in one more success, and allow for rolling two more dice.
4 combinations which result in two more successes.
4 combinations which result in two more successes, and allow for rolling two more dice.
1 combination which results in two more successes, and allows for rolling four more dice.

And that's as far as I got. Maybe I just need to math-hammer this some more, but I was hoping anyone else might see an intuitive solution.
 

steenan

Adventurer
It's worth noting that this d8 with double-explode can get multiple successes from a single die.

The probability of getting at least one success is 37.5%, but the average number of successes from a single die (including exploding) is 0.5
 

Saelorn

Adventurer
The probability of getting at least one success is 37.5%, but the average number of successes from a single die (including exploding) is 0.5
Thanks! I assumed it would be close to that, so I wouldn't have to calculate out the effect of further explosions past the first, and running from that assumption led me to believe that it's probably true.

In case anyone is curious, my manual calculations gave the following (approximate) breakdown for each die rolled:

62.5% chance of zero successes
29.9% chance of one success
3.9% chance of two successes
2.7% chance of three successes
0.8% chance of four successes
0.2% chance of five successes
 

DMMike

Game Masticator
That stuff gets complicated fast, huh? I bet superhero-type games would benefit from exploding dice that can reach chain-reaction. Other games, not so much.

XP for inspiring me to write up the Excellent Move (double action) and the Multi (square damage), in addition to exploding dice.

Excellent move:
Dragon-Quest-Monsters-Game-Boy-Color-Pixel-Art-Xtreme-Retro-Square-Enix.png
 

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