Dice for Non-Mathematicians - 201

Rolling At Leasts
Let's imagine a game where you I roll 1d6 and you roll 1d6 and have to tie or beat me.  So, I roll a 5. Sweet. What were the odds that I would roll a 5? Well, obviously 1 in 6.

Now, what are the odds that you will tie or beat me? Your possible roll table is [1,2,3,4,5,6], meaning out of 6 possibilities, you have 2 chances. Ergo, your odds are 2-in-6 (33%).

You probably could have worked that out without a roll table. What you probably didn't do, but is at the heart of this topic, is to add together the individual chances for each winning roll... but that's exactly how it works.



Now, let's kick it up a level. I roll 1d6 and you roll 2d4.

I roll a 5 again. What were the odds of that?! Actually, still 1 in 6, because my die doesn't remember what I rolled last time. Every single roll has the same probability.

So, with your 2d4, what are the odds that you will tie or beat me? Let's do a roll table.

1,1 = 2    2,1 = 3    3,1 = 4    4,1 = 5
1,2 = 3    2,2 = 4    3,2 = 5    4,2 = 6
1,3 = 4    2,3 = 5    3,3 = 6    4,3 = 7
1,4 = 5    2,4 = 6    3,4 = 7    4,4 = 8

You have a 10-in-16 (62.5%) chance of tying or beating my 5.

Now, you could also calculate this by figuring the odds of one outcome at a time and add them up.
You have a 4-in-16 (25%) chance of a 5.  You have a 3-in-16 (18.75%) chance of a 6.  2-in-16 chance of a 7 (12.5%) and a 1-in-16 (6.25%) chance of an 8.

(4+3+2+1)-in-16 chances.

So while the roll table of 1d4 will look like a bell, which is fat in the middle and thin on the ends, if you were looking at "At Least" values it would look like a curve, fattest on low end sloping down to a thin sliver.

That Second Roll
But let's say we didn't roll our 2d4 at the same time. Let's say we roll one, and then roll the other. We can pause midway through and update our chances for success.

So, if your first 1d4 comes up as a 1, you know the only way to score is by rolling a 4 with the second. So in between rolls, your probability of scoring just dropped from 62.5% down to 25%!

But if your first 1d4 came up as a 4, you would have a 100% success rate. No matter what you roll on the second toss, you are guaranteed to total at least 5.

Rolling Doubles
On a related point, many games use special mechanics around rolling "doubles" (two of the same number). Rolling doubles seems like a magical thing. While I don't want to strip away the mystique, I do want to demystify it a bit.

Say I'm rolling 2d4 (as outlined above). There are 16 possible outcomes... you can see them all listed there. What are the odds of rolling double 4s? 1-in-16 (after all, there is only one path to get you there... a 4 and a 4). The odds of double 2s? 1-in-16.

What about the odds of ANY double? Well, with 16 possible rolls, and 4 different possible doubles, you'd have a 4-in-16 (25%) chance.

If you want a couple quick formulas for figuring doubles out of two dice, you can use this:

One Specific Double = 1 / (Number of Sides on first die * Number of Sides on second die)
Example: Double 3s using 2d12 = 1/(12*12) or 1/144 (less than 1%)

ANY Doubles = [Number of sides on the SMALLEST die] / (Number of Sides on the First Die * Number of Sides on the Second Die)
Example: Any doubles 1d4 and 1d6 = 4/(4*6) or 1/6 (about 17%)

But remember, your dice have no memory. They don't know or care what the other dice say. They are dice.

So if you are rolling 2d6, the odds of rolling double 3s is 1-in-36. But if you roll your 2d6 one at a time and the first one comes up a 3 then the odds change to 1-in-6. Rolling the first 3 doesn't get you "halfway there." Statistically, it gets you way closer to being "there."


Adding Dice
So, let's say we're rolling 3d4 looking for doubles.  The number of possible rolls is pretty easy to figure:  4sides*4sides*4sides (total of 64).  I've drawn the roll table below.

1,1,1   2,1,1   3,1,1   4,1,1
1,1,2   2,1,2   3,1,2   4,1,2
1,1,3   2,1,3   3,1,3   4,1,3
1,1,4   2,1,4   3,1,4   4,1,4

1,2,1   2,2,1   3,2,1   4,2,1
1,2,2   2,2,2   3,2,2   4,2,2
1,2,3   2,2,3   3,2,3   4,2,3
1,2,4   2,2,4   3,2,4   4,2,4

1,3,1   2,3,1   3,3,1   4,3,1
1,3,2   2,3,2   3,3,2   4,3,2
1,3,3   2,3,3   3,3,3   4,3,3
1,3,4   2,3,4   3,3,4   4,3,4

1,4,1   2,4,1   3,4,1   4,4,1
1,4,2   2,4,2   3,4,2   4,4,2
1,4,3   2,4,3   3,4,3   4,4,3
1,4,4   2,4,4   3,4,4   4,4,4

If you count all of them, you'll find that you have a 40-in-64 chance to roll doubles.  Notice also, that each column has similar blocks: 1 block with four doubles 4, and 3 blocks with 2.  The reason why is that we don't care what number doubles is rolled.  Because you have an equal chance of rolling double 1s and double 4s, you could figure the whole table just by working out double 1s.  (10-in-16 is the same as 40-in-64).

So by introducing a 3rd die, our chances of rolling doubles goes way up.  In fact, it is more likely that you will roll doubles than that you won't.

Bottomline:
Rolling doubles may feel like magic, but setting out the roll table on paper can help give a clear view of how rare they really are.

Moreover, you can gain fine-grained control over them by adjusting the number of dice and which doubles "count" (and how valuable each type is).

Next week we'll look at the effects of re-rolling and locking dice in Dice for Non-Mathematicians - 301.