The Monday Post: D&D Dice Probabilities

So I was playing my weekly game this week of Dungeons and Dragons when I was presented a rather odd opportunity for my character: I Cleric was (literally) flying overhead of my character about 30 feet (thanks to a potion of flight) and the best my fighter could think of to do was throw his 2-Handed Bastard sword up like a spear to try to hit the cleric. I needed a natural 20 to hit. I got an 18. I missed, but not by much.


Not surprising, really. I only had a 5% chance to make it (1 in 20). This got me to remember about an odd little equation I use to figure out what the most likely roll for several dice will be. By this I mean if I have 2 standard dice the most likely roll is 7, right? There's a reason for that.

When you look at an even number die (other's exist), you have to split between 2 numbers to have even results: On a standard die that's between 3 and 4. If we wanted an average between those numbers we'd have 3.5, right? This same mathematics holds true for other kinds of dice: d8s average 4.5, d10s give you 10.5, and so on. Now, if we take multiple of the same die we just add these numbers together. Taking the standard 6 sider as example 2 of them give an average of 7. These two facts (the divide of one die and adding all of these dividends together) turns into the magic (if slightly unfriendly) equation of:

X * ((Y / 2) + .5)

where x and y are XdY,

This seems stupid, but this comes in handy when you want to guess at how long it might take to beat up a monster. Say you take my lovely sword from the opening. It hits for 2d4 against most opponents, though larger ones get 2d8 when I connect. That means, on average I do 5 damage, 9 against something like a giant (or really anything bigger than me). 

Now enchantments change things. They affect both your hit rate AND damage. So a 2d4 + 2 (the actual stat of my sword, since it is enchanted) means I deal 7 damage on average (5 for normal dice rolls, then add the +2). So the real equation becomes (in a little different style):

For XdY + Z, average damage is:

X * ((Y/2) + .5)

Yes, this is seriously what I think about from time to time when not doing homework. See you next week.