According to me, probability is the chance that something will happen.  Look it up in a dictionary and you may find three or four other definitions.  If you play a tabletop RPG, then you should have some bit of understanding of probability.  Because dice are involved, probability is involved.

If you get onto the forums, you can find some pretty in depth discussions of the topic.  Not surprisingly, Dungeons and Dragons attracts some pretty bright mathematical minds.  Actual statisticians and mathematicians play the game and in their off time they go through studies to determine various outcomes of various theoretical D&D situations.

For the average player this level of discussion is not necessary.  However, some understanding of probability is necessary.

For example, what is the difference in damage between something that does 1d8 damage and something that does 2d4 damage?  On average they will both do 4 points of damage per hit.  However, the item that does 2d4 damage will never do only 1 point of damage.  This means that in general your lower damage limit will be higher using the 2d4 damage.  But what about maximum damage?

This is where probability does come into play.  For something that does 1d8 damage, the probability that you will get max damage is 1 chance in 8 (12.5%).  For something that does 2d4 damage, the probability that you will get maximum damage is 1 chance in 4 times 1 chance in 4 or .25 x .25 or 6.5%.   Because you are rolling more dice, the chance that both dice will come up at maximum is actually less.  The trade off becomes dealing slightly more damage more consistently with less chance of doing maximum damage.

The concept here is to pay attention to what is actually going on when you are rolling all those dice.

Two key principles to watch are the law of averages and the gambler’s folly.

The law of averages says that the more dice you roll the more likely you are to get the average value of those dice.  This means that a 10d6 fireball is more likely to get 30 as its result than anything else and that it will be rare to find anything more than 40 or less than 20 as a result.  It also means that the difference between a 10d6 fireball and an 11d6 fireball is 3 points of damage, not 6.  Take it up to 20 dice and you start to get closer to that average number even more consistently.

This also means that over time the damage that your character deals will also average out.  If your character wielding her longsword is going to make an average of 50 damage rolls per level (10 encounters with an average of 5 hits per encounter) then he or she is going to do 50d8 damage during that level.  There are enough dice here to fully invoke the law of averages, so the character will do about 200 points of damage – not much more and not much less.  Kick this up to a Battleaxe and you’ll do another 50 points of damage during that level.

The gambler’s folly is nearly the opposite of the law of averages.  It says that each roll of the die is independent of all other dice rolls.  So just because you rolled 3 20’s in a row doesn’t affect the chance that you are going to roll a 20.  In essence it says that all the superstitions about dice rolling are just bunk, but I’ll leave that up to you to decide.

The hang-up that players get caught in is when they try to circumvent probability with probability.  Doing more damage by adding more dice is deceptive.  Doing more damage by adding bonuses – by eliminating probability is generally better.  Players read the PHB and when they see an entry like 2d6 they immediately think 12 damage.  The way they should read it is 6 damage with an equal chance of doing 2 or 12 damage.  Only when they roll a 7 damage 6d6 fireball (yep, it happened at my table) do they realize that the odds swing both ways.

On average something that does 1d4+2 will do more damage than a regular old longsword (1d8).  Though it will never do 8 points of damage, it will always do at least 3 and usually 4 damage, and the odds of doing maximum damage are 25% instead of 12.5% so it may not be a bad choice.

Math behind probabilities of chances to hit are a bit more complex, but only because you have to also understand the probabilities of different opponents having different defenses.

What does this all mean, and why should you care?  It means that if you want to get the most from your character you need to look at more than just the numbers in the book.  Think about what they actually mean during play and what they will mean during session ten and twenty.

Or just skip the math and have fun!

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