The Average Dice Roll
Determining the average roll for any normal die is easy. Simply take the number of sides the die has, cut it in half, and add .5 to the total. We get this one of two ways. You can either average the highest and lowest result together (i.e. 1D10 would be (1+10)/2) or, even better, average all the possible results together (i.e. 1d4 would be (1+2+3+4)/4). Either way you should get the same result, unless you have a die with odd faces like a D4 that has a 1, 20, 3, and 55 for possible results.
With this, you should get the following:
- 1d4 averages to 2.5
- 1d6 averages to 3.5
- 1d8 averages to 4.5
- 1d10 averages to 5.5
- 1d12 averages to 6.5
- 1d20 averages to 10.5
- 1d100 averages to 50.5
Factoring in a static modifier is as easy as adding the bonus/penalty to the average roll. If your PC in D&D has a +3 strength bonus, then on average they will roll a 13.5 when making Strength checks. A character with a +7 to Open Lock checks will, on average, be able to open locks of difficulty 17.5 or lower.
Rounding To Integer
Mathematically a fraction or decimal of .5 would round up. In most RPGs however we round down. This means that your average 2.5 on a D4 is going to be represented in most cases as a 2 instead of a 3. How you personally factor this is up to you.
That said, there is one thing to note. When you are going to round a pool of dice, always wait until after the rest of the match to do the rounding. So if someone has to roll 3d6 do the addition (result = 10.5) before deciding what to do with the .5. Otherwise you could end up with expecting 2d6 to do 8 damage on the regular, or 6, when in truth it is going to do 7 and that extra point either way can be enough to mess everything up.
More than averages, probability tends to be the bigger thing most players try to factor. We all know the odds of getting a 6 on 1d6 is 1/6, but what about in other ways? What is the probability that Sarah's Thief can actually hit the BBEG to deliver her massive sneak attack? How frequently can you expect the Captain of the Guard to hit Barry's Paladin? What are the chances of a PC getting two natural 20s to make a set piece encounter look like you set the dial to "Super Easy" for the encounter?
There are tons of websites that will go over the specific probability any pool of dice will have of generating specific results. I'm not interested in those as much as I am in the more useful quick and dirty probability. That said, here are some of those.
On Average = 55% Of The Time or Better
For the most part when something is the "average roll" that means it will happen about 55% of the time. The average roll of a D20 is 10.5, rounded down and you get a 10. There is a 55% chance to get a 10 or better on a D20.
This math is useful just for determining if something is likely to succeed or not. The more above or below the average result, the more/less likely it is to happen.
Combine this knowledge with the difficulty scaling your system has and you can see just how badass your PCs are. Most systems don't take too long before the PCs have a 55% chance or better at succeeding against Hard and Very Hard difficulties. I like to celebrate when that happens with a session or two that shows just how good the PCs are. Let them know they've entered a new world of power.
D20 = 5% Increments
Each face on a D20 has a 5% chance of coming up. So if you take the number of facesF (results) that will end up in success and multiply it by 5 you will have the % chance a character has of hitting. In games like D&D 5th Ed this is carefully controlled as a part of game balance, while in other games the PCs start leaving the importance of the D20 itself behind as they get stronger and stronger.
Still, if Barry has an AC of 18, and your BBEG has a +12 bonus to his attack roll, then any result of 6 or higher is going to hit Barry. That means Barry gets hit 70% of the time, or out of 10 rounds Barry can expect to be hit 7 times (assuming one attack per round.) Combine that with the attack's average damage roll (let's say 10 damage) against Barry's HP total (45) and Barry will be down, unless he gets help, after 5 attacks.
Probability and Strings of Rolls
One of the fun things with probability is that it changes from moment to moment, and you need to be aware of this as a GM or player involved with a hobby based around dice.
For example, what are the odds of rolling consecutive natural 20s? Well, before the roll it is 1 in 400. Of the 20 possible results for the first die only 1 is acceptable (so a 1/20 chance) and of the 20 possible results for the second roll only 1 is acceptable. As such you have a 1 in 400 chance (1/20 * 1/20 = 1/400). However, let's say the first die comes up 20. What are your chances now?
Here's a hint, it's not 1/400 anymore. No, once the first die hits the table and stops rolling your odds are either 0 (you didn't get the first 20) or 1/20. Why? Because 380 potential results have just gone out the window with that first die settling with a face up.
Why is this important? Mostly because it combats dice superstition (though people will still believe it) but also just to curb your expectations. Is it uncanny for a D6 to roll 12 6s in a row? Sure. That's very unlikely to happen. Of course, if the D6 has already rolled 6 11 times in a row, that 12th 6 is just a 1/6 chance away. All the other failure conditions you had at the beginning are gone.
The Only Things You Need To Know
Years ago I read a book called "Improbable" about someone who received the ability to see the branching future paths (a gambler) and figure out how to make things happen. The book was fun but it also had 2 lines that stuck with me and changed my view on probability for the better.
The first is this: the only thing you need to know about low probability events is that sometimes shit happens. The idea being that you shouldn't be too surprised about low probability events happening because - eventually - they do have to happen, and sometimes you're just the unlucky person who catches it. Don't get caught up in shcok, just deal and move on.
The second helps with that. I don't remembe the exact quote but it boils down to: As long as I can be sure the probability of something happening is better than 1 in 7 billion, and can be reasonably sure that other people in the world aren't experiencing that exact thing at that exact moment, it's not all that surprising that it is happening to me.
The second quote is basically the same as the first. But the idea is there are 7 billion people in the world. That means that at any given moment something that is a 1 in 7 billion chance of happening should be happening to someone. So why not you?