 Can Someone Confirm This Math  

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The Mad Hatter Hero
Posts : 26 Trading Reputation : 0 Join date : 20120721 Age : 42 Location : San Jose, CA
 Subject: Can Someone Confirm This Math Fri 24 Feb 2017  23:16  
 OK, as opposed to tacking this onto the already extremely long double wield thread, I wanted to just get a sanity check.
What I'm looking to test: Chances of wounding in two weapons vs weapon and shield scenario given the two rules below.
Rules used: Hand weapon and shield rule (from 7th ed Warhammer) which gives a user with hand weapon and shield a 5+ save in close combat (6+ save in ranged). Fighting with two weapons is done at 1 WS for a dual weapon wielder
Assuming all characteristics standard and the same for both combatants: WS S T A W I 3 3 3 1 1 3
When I run the numbers, I get the chance to cause an unsaved wound to be exactly the same at .33. To me this tells me that ignoring all other special rules, using the hand weapon and shield rule from 7th edition Warhammer, against the 1 WS house rule for dual wielding gives the exact same percentage chance to cause a wound (thereby making the choice between HW and shield versus dual wielding exactly the same in terms of chances of causing a wound for two baseline stats opponents with no other special rules).
Now tacking on other special rules can change the outcome, but I'm looking to understand with a baseline of two equal warriors, using the above two rules, you have exactly equal chances in a fight. If you run the scenario, does your math produce the same result? 

 
Athanatosz General
Posts : 175 Trading Reputation : 0 Join date : 20160209 Age : 32 Location : Hungary
Personal Info Primary Warband played: Marauders (BTB) Achievements earned: none
 Subject: Re: Can Someone Confirm This Math Fri 24 Feb 2017  23:35  
 I'm not that got at math but i have a question: whats are the chances if you only increase the WS attribute to 4 and leave all the other in the same? 

 
The Mad Hatter Hero
Posts : 26 Trading Reputation : 0 Join date : 20120721 Age : 42 Location : San Jose, CA
 Subject: Re: Can Someone Confirm This Math Fri 24 Feb 2017  23:41  
 If you raise both WS to 4, and then apply the 1 WS to the dual wielder (giving him a final WS of 3), the percentages work out exactly the same  .33 chance to cause a wound for both.
If you were only to increase the weapon and shield guy to 4, giving you a WS 4 vs 2, it still remains .33 for both.
If you only raise the dual wielder to 4, giving you a WS 3 vs 3, your percentages change to .25 for weapon and shield to cause a wound, and .33 for dual wielder to cause a wound. 

 
mitokun Warrior
Posts : 17 Trading Reputation : 0 Join date : 20160214
Personal Info Primary Warband played: Marienburgers Achievements earned: none
 Subject: Re: Can Someone Confirm This Math Sat 25 Feb 2017  7:30  
 SW attacking DW:
Chance to hit: 4/6 Chance to wound: 3/6
Altogether: (4/6)*(3/6) = 12/36
DW attacking SW:
Consider one of his attacks:
Chance to hit: 3/6 Chance to wound: 3/6 Chance opponent failes his save: 4/6
Altogether: (3/6)*(3/6)*(4/6) = 1/6
Therefore the chance both his attacks fail: (5/6)*(5/6) = 25/36 Therefore the chance that at least one attack goes through: 11/36
More interesting (in my opinion) is the expected damage done: (1/6)+(1/6) = 12/36
Result:
Yes, I can confirm that the chance that at least one attack goes through is in both cases roughly 0.33.
Although the chances that at least one attack goes through are slightly different, 12/36 and 11/36, the expected damage done is exactly the same with 12/36 each. 

 
The Mad Hatter Hero
Posts : 26 Trading Reputation : 0 Join date : 20120721 Age : 42 Location : San Jose, CA
 Subject: Re: Can Someone Confirm This Math Sat 25 Feb 2017  8:12  
 That's very interesting that the chance of attack going through is slightly less (12/36 vs. 11/36) yet the chance of actually doing damage is exactly the same at 1/3?
I'd thought because the chance of doing damage is exactly the same at 1/3, you could make the statement that dual wielding is no better or worse in combat than HW and shield.
In reality, the chance you cause damage is exactly the same, but the actual chance of you hitting is just slightly less......So what should we really care about, that the chance of actually causing damage is the same or the chance that you actually hit is the same if the goal is to "even out dual wielding"?
In the end, would this change anyone'e perspective about using all dual wielding by default?
My personal thinking is that the advantage of dual wielding would then come down to the fact that you can wield two different weapons with different special rules (i.e. a sword to benefit from parry and an axe to punch through armor). This could provide you a slight advantage in combat, whereas the HW and shield armed model benefits from a better save at ranged since he's carrying the shield, while having the increased save in hand to hand (which as we've sort of shown, balances out against a dual wielding opponent in terms of base chance to wound with no special rules being considered).
Any thoughts?


 
mitokun Warrior
Posts : 17 Trading Reputation : 0 Join date : 20160214
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 Subject: Re: Can Someone Confirm This Math Sat 25 Feb 2017  9:08  
  The Mad Hatter wrote:
 yet the chance of actually doing damage is exactly the same at 1/3?
No, the expected value is 1/3. The 'expected value' can be used to answer a question like this: "How many wounds is my model expected to cause if it attacks 6 times?" (answer: "Two wounds.") Note that the above question is hard to answer if we keep talking about 'chances'.  Quote :
 So what should we really care about, that the chance of actually causing damage is the same or the chance that you actually hit is the same if the goal is to "even out dual wielding"?
What we should care about is the expected value of the actual damage done which enables us to make reasonable comparisons. For example you can take a look at the change of the expected value when adding +1 A or +1 S. 

 
The Mad Hatter Hero
Posts : 26 Trading Reputation : 0 Join date : 20120721 Age : 42 Location : San Jose, CA
 Subject: Re: Can Someone Confirm This Math Sat 25 Feb 2017  10:21  
 Well right now, the chances of damage done is equal so by that comparison, you could say all things being equal for two base line characters, with the two rules note above, dual wielding and HW and shield are by all terms equal.
I can definitely play with increasing other things, A or S as examples, you've note, and see how that changes things. My thought is increase A for both sides, see chances. Increase S for both sides, see chances, are those your thoughts as well? 

 
The Mad Hatter Hero
Posts : 26 Trading Reputation : 0 Join date : 20120721 Age : 42 Location : San Jose, CA
 Subject: Re: Can Someone Confirm This Math Sat 25 Feb 2017  10:30  
 OK, increasing S to 4 for both sides gives this result for chance to wound:
DW: .56, HW&S: .44
Increasing base A value to 2 for both sides gives this result for chance to wound?
DW: .5, HW&S: .67
Increasing both A value to 2 and S to 4 for both combatants results in:
DW: .83, HW&S: .89
I also ran a few more scenarios, increase T, add light armor to both....Seems to maybe have slightly skewed results into HW&S being favorable. I'm not sure we'll ever get total balance, but looking for the right combination of rules that gets very close balance. I'm not sure if I'd classify this as very close or not, but definitely better than the starting point with existing rules as written. For reference, chance to wound with existing rules as written with baseline stats are:
DW: .42, HW&S: .25 

 
Athanatosz General
Posts : 175 Trading Reputation : 0 Join date : 20160209 Age : 32 Location : Hungary
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 Subject: Re: Can Someone Confirm This Math Sat 25 Feb 2017  11:47  
 And what if we considers the chances that your Armour save will work? Also hit chances count attacks from the second weapons? 

 
The Mad Hatter Hero
Posts : 26 Trading Reputation : 0 Join date : 20120721 Age : 42 Location : San Jose, CA
 Subject: Re: Can Someone Confirm This Math Sat 25 Feb 2017  15:57  
 If by chances of your armor save working you mean accounting for the armor save, then yes, that's been factored in. So has the chances of scoring additional hits from the second weapons.
The only thing that hasn't been accounted for is any particular special weapon rules (i.e. if the weapons happen to be swords or axes, I haven't taken into consideration that you may get a parry save or be 1 to an armor save). This is meant to generalize probabilities, not take into consideration the exact weapon special abilities. Being armed with a sword and dagger for example would actually have a different outcome than say a sword and axe.
I want to generalize it to get a baseline, as adding in all the complexities from different weapon rules makes it a much more complex distribution. 

 
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 Can Someone Confirm This Math  
