## Exploding dice

My Dungeon Master (DM) was worried. The fights were getting a bit lengthy, or as they say, “grindy”. There were 2 ways to speed things up. One, to make it easier to hit. The other, was to make any hits more damaging.

[image by heizfrosch]

So my DM used the exploding dice concept. Basically, for any damage die roll, if you roll the maximum, you get to roll that die one more time. In theory, you could keep rolling dice till the penguins flew. Oh wait a minute…

Common sense says that, the more dice you get to roll, the higher chance of you rolling an exploded die. Thus 4 six-sided dice (hence denoted by the short form 4d6) is better than 2 12-sided dice, even though their totals are the same.

I think there’s also a probability result that says the more samples you get, the closer to the mean you’ll be. Thus 4d6 is “safer” than 2d12, because an average of 14 from 4d6 is better than a 2 from seriously flawed rolls of 2d12.

[corrected average value of 4d6 from 12 to 14. Thanks to ugasoft for pointing it out.]

Anyway, I’ve played with my group using this exploding dice concept for some time now. From personal experience, it speeds up combat somewhat erratically. What I mean is, one game battle could drag for some time, while another has us taking the monsters down quickly. I’m playing a character that deals more dice rolls than others, yet I don’t seem to roll exploding dice often. Just roll a critical, a 20 on a d20 (does maximum damage, no need for die rolls). Much faster that way.

Anyway, I wanted to do some mathematical analysis on this exploding dice concept. You can probably tell from the above that I probably know very little about probability. Well, I hated statistics in university.

I was trying to come up with some lame math formula, then I had the brilliant idea of searching on the Internet. And I found Eric, who did an actual math analysis of exploding dice. Much better than my feeble attempt. I almost wanted to tear up all my calculations in shame. His conclusion:

For any N-sided die numbered 1 to N with all sides equally likely, the exploding modifier will increase the die’s expected value by a factor of N/(N-1)

Reading his article, some faint memory came back to me. Expected value, huh? So the expected value of a d6 is
(1+2+3+4+5+6) / 6 = 3.5

And the expected value of a d12 is
(1+2+3+4+5+6+7+8+9+10+11+12) / 12 = 6.5

So the expected value of 4d6 is 4 * 3.5 = 14,
and the expected value of 2d12 is 2 * 6.5 = 13

Even without Eric’s conclusion, we can see 4d6 is better than 2d12, but let’s finish the calculations. Expected value after explosion for 4d6 is 14 * 6/5 = 16.8, and expected value after explosion for 2d12 is 13 * 12/11 = 14.18…

The takeaway? If you have to choose between 3d6 and 2d10, go for the 3d6, even though 18 (3 * 6) is less than 20 (2 * 10). You’re more likely to roll more stable higher results. Knowing this does nothing for my unlucky rolls of 1’s though…

## I suck at rolling dice

I really do. And I’ve a story to prove it. The origins are a bit complicated, since it involves a friend of a friend, who introduced me to another friend. It also involves a blog I read, an RSS feed and a series of podcasts indeed (hey it rhymes!).

I read Scott Beale’s blog, Laughing Squid. Can’t remember why I started reading it… Anyway, he posted something by Wil Wheaton. Now, I have no idea who Wil was, and after some reading of Wil’s posts (and some Googling), I found that he’s actually kinda famous. Wow.

Now Wil was writing about his experiences with Dungeons and Dragons (apparently he needs to save vs shiny when he goes to the local game shop). And I found out that he’s in a podcast, collaborating with Penny Arcade and PvP. I learnt fun facts about Jim’s Magic Missile and Rudy the Undead Hound (uh Ru-ru-ru-ru-ru-dy!).

So I Plurked about it. And a friend (originally a friend of a friend, but let’s not go there…) replied that he knew a friend who’s a Dungeon Master. And that’s how I found my first Dungeon Master. And I had my first game during that break of mine. Wait, there’s more…

I only knew there were 2 other players. So in total there are 3, including that Dungeon Master. One of the players expressed interest in playing a striker (meaning he deals obscene amounts of damage to one enemy at a time). I bought the Player’s Handbook, and after some deduction on the roles, concluded that the other player was probably a cleric of sorts, so there’s some healing capability. From the podcasts I listened, healing is extremely important. Enemies can seemingly fell characters without so much as breaking a sweat…

On the way to the appointed place, on a whim, I Plurked about it. That mutual friend replied, saying to have fun. And I arrived at the meeting place. And a funny sinking feeling settled in. I have no idea how the Dungeon Master friend and the other 2 players look like! I don’t have their phone numbers too. The only contact information I have was Facebook and email addresses. Oops.

Now, it happened that I tied my Plurks to Facebook updates. So I replied to the Facebook update, hoping that mutual friend was still online. He was. Whew. After some exchange of replies, I gave him my phone number via a private Plurk (yes, I don’t even have his number, nor does he have mine), and asked him to send it to that Dungeon Master friend. In the end, I managed to meet up. The power of 2 social media sites working together…

Remember I deduced the roles of my fellow players? I was wrong. Well, half wrong. There was a striker and a leader (meaning buffing stats and some healing and stuff). But the leader role wasn’t a cleric. He was a warlord. You don’t have to know what it means in game terms… What it does mean is that both players are offensive. Which makes sense, because with only 2 characters, your best defense is all out offense.

My mistake was creating a relatively well balanced character (an eladrin wizard), able to fight in battles and social encounters alike. It also means I’m not maximising my damage, which was debilitating, because I was less useful to the group. It was a one-off encounter, so the characters aren’t meant to be used in future encounters.

It doesn’t help that I miss half the time too.

I’ve never rolled d20 dice before. That’s a 20-sided die, and it’s an awesome polyhedron to look at. I thought I should have beginner’s luck or something. Nope. One of the players said I have beginner’s unluck…

From the podcasts, the dice rolls were fairly varied. There was also one critical moment, when someone got to roll another attack. His first attack missed critically (he rolled a 1 on d20). So he got another chance (he used an Action Point), and rolled … another 1. That’s a 1 out of 400 chance of consecutive critical failures happening. It was hilarious.

Never did it occur to me that I’ll face a worse fate…

There I was struggling to keep alive. I was poisoned, and kept barely alive by a fellow comrade. I cast the famous Magic Missile, and it missed. Critically. A 1 on a d20.

Ok then. I use an Action Point to gain another action. I cast a fire sphere spell, and it missed. Critically. Another 1 on a d20. What?!?! (remembers “Is that a 1?” from the podcasts).

I was defeated. I apologised to my comrades for my ineptitude. They said it’s fine. The good thing was, even though my fire sphere missed, the fire sphere was created, and it would do damage if an enemy was beside it. So not a total loss, but still…

Well, it’s the end of my turn, so I do a save versus poison roll. And I rolled a … 1. On a d20. Not only did I fail to recover from poison, I failed epically. Great. I didn’t know whether to laugh or cry.

My comrades were all run-down, and one of them had to continually heal me, because if I die, the fire sphere disappears. I feel so useless. The fire sphere does more damage in total than if I hit the enemy in the first place.

Well, my turn came back again. Since the fire sphere was already in place, I gave it another shot. I didn’t have any dice of mine, so I used other people’s dice. I didn’t realise it, but I kept using the same d20. It didn’t matter to me then, so I just picked it up, rolled it in my hand, and let it drop onto the table…

And a 1 came up to greet me.

That d20 really hate me.

I had failed critically, by rolling 4 consecutive 1’s on a d20. That’s a 1 out of 160000 chance of it happening. I suck at rolling dice…

I can’t remember much after that. I rolled to save against my ongoing poison, and failed. Not critically, but I still failed. After my turn, it was the monster’s turn. I jokingly told the Dungeon Master to use that “cursed” die. He did, and rolled a 1. Ha!

At that point, the other players and I convened, and decided that we’re way out of our league. All our resources were expended. The monster was still going strong. And did I mention that we’re dying?

We retreated up the ladder a bit, keeping close enough so I could maintain my fire sphere spell. The Dungeon Master was kind enough to let the monster get roasted alive. We returned to the town to rest, and came back to finish off the rest of the adventure (which wasn’t much).

We finished off by doing a recap. I was still bemoaning about my die rolls, and the Dungeon Master gave me this piece of advice.

Don’t rely on luck.