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This post is an analysis of the probability of the chances for obtaining a particular result of a dice roll for the White Wolf role playing game system. If you're already familiar with the system, you can probably skip ahead to the "Read More" link below and look and the graphs I've provided. If you're not well versed in the mechanics of the White Wolf role playing game system, please keep reading.
In the role playing game system developed by White Wolf, the success or failure of characters' actions is determined by the number of successes on a given dice roll. All rolls use ten-sided dice and the total number of dice thrown, or dice pool, is determined by the sum of a character's rank in a primary attribute and a secondary skill. The number of successes is determined by the number of dice rolled with a value equal to or above the target difficult. A typical difficulty is seven and easy actions will have a lower difficulty while more difficult actions will have a higher difficulty. The minimum difficulty is three and the maximum difficulty is ten.
For a simple action, only one success is needed is successfully complete that action. An example of a simple action would be trying to fire a gun and hit a target. This is a straight forward act and is neither exceptionally easy nor exceptionally difficulty. Thus, it would have the standard target difficulty of seven. A character performing this action would have a total dice pool equal to the sum of their ranks in the dexterity attribute and the firearms skill. For example, a character with a dexterity rating of three and a firearms rating of two would have a total dice pool of five allowing them to throw five dice to determine if the succeed or fail at the action. If they were to roll a 2,5,8,4, and 6, they would have acquired one success since only one die, the one showing eight, had a value greater than or equal to the target difficulty of seven. In that case, it would be up to the Storyteller to determine the effect of only a single success. While the character will have hit the target, it may have been only a glancing blow. If the character had rolled 4,7,9,10,7 and gained four successes, the Storyteller may interpret the greater number of successes as a superior result and say that the character hit the target dead center. A roll of 2,5,6,4,3 would net zero successes and represent a miss since none of the dice rolled showed a value greater than or equal to the target difficulty of seven.
In the White Wolf success system there is the “rule of one” and there is also the critical success rule. The “rule of one” states than any die that shows a one cancels out a single success. So, if a character rolled 1,3,7,8,5 with a target difficulty of 7, one of the two successes would be canceled out by the one resulting in only a single net success. On the other side of the coin, so to speak, is the critical success roll. If the character has a specialty – an area of focus chosen for skills that reach rank four or higher – any ten that's rolled and scores a success may be rolled again for a chance at additional successes.
The discussion of the “rule of one” leads us to a discussion of critical failures or botches. White Wolf has had different incarnations of the botch system that we'll split up into what I'll call the old botch rule and the new botch rule. Under the old botch rule, if you rolled more ones than you did successes – i.e. you scored a negative number of net successes – you rolled a critical failure which the Storyteller is encouraged to interpret as creatively as possible. Thus, if our previously mentioned character rolled a 1,3,1,7,4 with a target difficulty of seven, they would botch their action which the Storyteller could interpret as anything from the character's gun jamming to the bullet sailing high over the target to strike the character's grandmother that he didn't see up over a ridge downrange. Under the new botch rule, the character only botches if they roll a one and no successes. For example, if our character rolled a 1,1,4,2,6 with a target difficulty of seven, they would botch under the new botch rule, but the would not botch if they rolled a 1,1,4,2,7 with a target difficulty of seven despite the fact that they rolled more ones that successes. As you can see, the new botch rule is a little more forgiving than the old botch rule.
Now, as just about anyone who has played a game involving dice can tell you, frequency statistics can be wildly skewed by small sample sizes and player perception. To aid players in having a more reasonable idea of what the probability of achieving a certain result is in the White Wolf system, I designed a numerical simulation where a random number generator was used to simulate 1,000,000 dice rolls for every combination of difficulties and dice pool sizes for the full range of difficulties of three through ten and a range of dice pool sizes from one to twenty. The curves in the plots that are to follow show the percentage chance of rolling a selected number or more successes for a given difficulty and a given sized dice pool.
The charts themselves are pretty self explanatory. Simply choose a chart based on whether you're playing under the new botch rules or the old botch rules and by your target difficulty. From there, select the minimum number of successes that you would like and use the number of dice in your dice pool to select the spot on the curve where you can obtain the probability of meeting or exceeding your desired minimum number of successes. For example, if you look at the chart for a difficulty of seven under the new botch rules, you'll see that for a dice pool of five you have about a 75% chance of rolling one or more successes, a roughly 20% chance of rolling zero successes, and approximately a 5% chance of botching the roll. Please click "Read More" to see the charts.
My friends and I, when playing White Wolf games, swore to ourselves that you had a greater chance to botch when you threw more dice. We never calculated out the probability and we debated amongst ourselves about rather or not that was the case and if it wasn't a counter intuitive result. If you examine the probability cures for the higher difficulties, you'll see that you do have a greater chance of botching your roll if you throw more dice, but only for fairly small dice pools. Usually, once your dice pool in above five, that's no longer the case. Also, note that for the old botch rules that for a difficulty of ten the probability to botch a roll is the exact same as the probability to roll one or more successes.
New Botch Rules
New Botch Rules - Difficulty 3
New Botch Rules - Difficulty 4
New Botch Rules - Difficulty 5
New Botch Rules - Difficulty 6
New Botch Rules - Difficulty 7
New Botch Rules - Difficulty 8
New Botch Rules - Difficulty 9
New Botch Rules - Difficulty 10
Old Botch Rules
Old Botch Rules - Difficulty 3
Old Botch Rules - Difficulty 4
Old Botch Rules - Difficulty 5
Old Botch Rules - Difficulty 6
Old Botch Rules - Difficulty 7
Old Botch Rules - Difficulty 8
Old Botch Rules - Difficulty 9
Old Botch Rules - Difficulty 10
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