One of the great things about playing warhammer fantasy is the ability to cast magic. Being a fickle force however, there is great risk to trying to tamper with the sorcerous ways. You may already know that this risk is represented by rolling a miscast, or two ones while casting a spell. I have noticed that some players will not risk rolling a lot of dice when casting a spell, the rationale is that they have more of a chance of rolling two ones. Is this true? Well, lets look at some probability to find out.
When rolling two six-sided dice, there are 36 possible combinations that can come up. To get this number you take the six combinations that can turn up for the first dice and multiply it by the six combinations that can arise from the second. 6 x 6 = 36
Out of those 36 possibilities, two ones can only happen 1 way! So, the probability of rolling two ones on two six-sided dice is 1 out of 36 or 0.03. See the chart below:
| | 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | 1,1 | | | | | |
| 2 | | | | | | |
| 3 | | | | | | |
| 4 | | | | | | |
| 5 | | | | | | |
| 6 | | | | | | |
Now, sometimes you need more dice to cast so that your spell will be successful. Let's see what the chance of rolling two or more one's is now. In order to miscast with 3 dice you need to roll one of the following combinations
| Die 1 | Die 2 | Die 3 |
| Roll 1 | Roll 1 | Not a 1 |
| Roll 1 | Not a 1 | Roll 1 |
| Not a 1 | Roll 1 | Roll 1 |
| Roll 1 | Roll 1 | Roll 1 |
This is four combinations. The total possible roll combination for three dice is 6 x 6 x 6 = 216. So, the new probability of rolling at least two ones is 4/216 = 0.018. This is SMALLER than the previous result. Let's do it one more time for four dice and see if the pattern continues.
| Die 1 | Die 2 | Die 3 | Die 4 |
| Roll 1 | Roll 1 | | |
| Roll 1 | | Roll 1 | |
| Roll 1 | | | Roll 1 |
| | Roll 1 | Roll 1 | |
| | Roll 1 | | Roll 1 |
| | | Roll 1 | Roll 1 |
| Roll 1 | Roll 1 | Roll 1 | |
| Roll 1 | Roll 1 | | Roll 1 |
| | Roll 1 | Roll 1 | Roll 1 |
| Roll 1 | | Roll 1 | Roll 1 |
| Roll 1 | Roll 1 | Roll 1 | Roll 1 |
There are 11 ways of rolling at least two ones now. There are 6 x 6 x 6 x 6 = 1296 combinations total. That means there is a 11/1296 = 0.0085 which is smaller than the two probabilities above. Already, I have debunked the myth that by rolling more dice you will have a greater chance of miscasting. For those who are mathematically inclined the formula for finding the probability based on a number of dice is:
(2n-nC0-nC1)/6n where n is the number of dice rolled and C is the Combinatorics function.
For the probabilities for rolling 2 to 10 dice, I have included them below in percentage form.
| number of dice | probability |
| 2 | 2.78% |
| 3 | 1.85% |
| 4 | 0.85% |
| 5 | 0.33% |
| 6 | 0.12% |
| 7 | 0.04% |
| 8 | 0.01% |
| 9 | 0.00% |
| 10 | 0.00% |
Hiya Ryan, interesting piece, but I'm afraid your maths is wrong :(
ReplyDeleteFor your '3 dice' table you should have an entry for each possible number on the 'Not 1' dice. For example; instead of row one saying 'Roll 1, Roll 1, Not a one' there should be five rows saying 'Roll 1, Roll 1, Roll 2' then 'Roll 1 Roll 1, Roll 3' then 'Roll 1, Roll 1, Roll 4' etc.
I don't have the correct formula to hand, but a 3 dice roll will give a probability of a miscast on about 0.07, rather than about 0.02 for two dice.
Although... it would be nice to roll 8 dice and only have a 1 in 10,000 chance of a miscast!
Your post failed the common sense test immediately!
ReplyDeleteObviously if you are rolling more dice there is more chances for 1s to be rolled not less...